This is one of the most crucial questions I have encountered on Philosophy.SE, and indicates an admirable consciousness about the structural bias in Western mainstream academia. It's a fact that while philosophy according to the Western narrative and experience is taught and studied almost all across the world, eastern wisdom/philosophies are much less widely studied in the West.
So for all the good reasons I decided to dedicate time to write a comprehensive answer to this, however it demanded delving into critical history of modern Western civilization which falls into disciplines such as Postcolonialist studies and criticism of the western Orientalist culture via critical discourse analysis, hence the length of my post and a large section of it having to touch upon these fields which are not directly relevant to the study of philosophy but the sociopolitical origins of Western philosophical discourse.
The Sociopolitical Factor
An important factor is the historically disadvantaged sociopolitical situation of Eastern countries who should've been, otherwise, promoting and representing their own philosophical/cultural heritage in the world.
It is important to note that over the last century most Asian countries have been under repressive political, economic and military dominance of Western powers such as the British, American or Communist *empires who have been always enforcing their own ideology and culture over the conquered nations. Ask yourself for example, why a lot of people in India speak English as their first language and are forgetting their own native tongue, or why Turks abandoned their own alphabet, etc. Examples and indications of the strong influence of Western colonialism over indigenous cultures are too obvious around the world and too numerous to list.
As a real example, my country Iran and its indigenous traditional culture went under the attacks of Western secular academia and culture during the reign of Western-backed secular Pahlavi Monarchy. A lot of our tradition sciences and wisdom (e.g. Islamic traditional medicine, religious seminaries as our traditional academic institutions, etc) were repressed and marginalized under the influence of colonialist interests that became state policy of Pahlavis. But thankfully there have been an academic resurgence of interest in our traditional thoughts following the glorious Islamic Revolution of 1979 since when Iranian traditional thinkers have also been making advances in active representation in world academia.
* Although Communism as a political government emerged in Russia, it owes itself entirely to the West, both in terms of theory (based on Marxism) and political success. For the evidence of the latter part read: Non Dare Call It Conspiracy by Garry Allen.
Western Cultural Attitude towards East
The other important factor is the phenomenon of Eurocentrism which have permeated all levels of modern Western civilization including academia. It's a general cultural mentality expressed:
... in terms of dualisms such as civilized/barbaric or advanced/backward,
developed/undeveloped, core/periphery, implying “evolutionary schemas
through which societies inevitably progress” supposedly with a remnant
of an “underlying presumption of a [supposedly] superior white Western
self as referent of analysis”
Following the so-called Enlightenment in 18th Century Europe and subsequently the Industrial Revolution underpinned by technological booms of the 19th century, Western man came to view itself as a superior, ‘advanced’ civilization over the traditional cultures who were lagging behind only in technological progress. This was while Eastern culture, specifically, great contributions of muslim philosophers and scientists during the Islamic Golden Age, had a major and consequential role in the rise of science in modern Europe, admitted by learned and fair-minded Orientalists such as Robert Briffault who wrote in his The Making of History:
[T]here is not a single aspect of European growth in which the decisive influence of Islamic Culture is not traceable, nowhere is it so clear and momentous as in the genesis of that power which constitutes the permanent distinctive force of the modern world, and the supreme source of its victory, natural science and the scientific spirit... The debt of our science to that of the Arabs does not consist in startling discoveries or revolutionary theories, science owes a great deal more to Arab culture, it owes its existence. (pp.190-191)
However, despite the fact that muslims’ approach to sciences was holistic and all sciences were regarded as a part of religious wisdom, in West, science was approached with a disregard to the unifying metaphysical principles that works to direct scientific endeavor towards transcendental truth both in practice and theory. (Closely relevant to this idea is my answer here to a question on quantitative vs. qualitative philosophy).
Eurocentrism was first scholarly examined by Edward Said, a Palestinian American thinker notable for his book Orientalism * in which he analyzes “the cultural representations that are the basis of Orientalism, a term he redefined to refer to the West's patronizing perceptions and depictions of Middle Eastern, Asian and North African societies—"the East".” Therefore the thesis of Orientalism is
the existence of a "subtle and persistent Eurocentric prejudice against Arabo-Islamic peoples and their culture", which derives from Western culture's long tradition of false and romanticized images of Asia, in general, and the Middle East, in particular. That such perceptions, and the consequent cultural representations, have served, and continue to serve, as implicit justifications for the colonial and imperialist ambitions of the European powers and of the U.S. Source
* A more detailed review of Said's Orientalism Thesis can be read here.
The Prevalent Epistemological Attitude of the West
Apart from these important sociopshycological factors, one can argue that philosophical thoughts of the European Enlightenment themselves had an important role in alienating Western Civilization from Eastern philosophies. Important developments in philosophical/epistemological thought during Renaissance and Enlightenment contributed to the formation of a general culture that shies away from religious, intuitive and mystical approaches to truth (important features of many eastern philosophies) and considers them to be dogmatic, unverifiable or mythical.
The Catholic theology that contradicted many scientific findings, and the failure of "Rationalists" such as Desecrate to successfully account for theological beliefs led to the Kantian conclusion that even metaphysics is beyond human cognition, which precipitated the alienation from intangible/transcendental realities culminating in the intellectual infatuation with Empiricism since early 20th century.
So your suggestion is to a large degree true that the reason eastern philosophies are not popularly discussed in western Academia is because of the general epistemological distaste for intuitive thoughts whose known proponents (to the Western academia at least, as we will see) have mainly discussed their ideas via descriptive imagery.
However, not only it is wrong to assume that descriptively-discussed intuitive thought is necessarily mythical or even unverifiable, it's also a typical Eurocentric ignorance that all Eastern philosophies have been exclusively discussed descriptively.
My Experience of the Largely Neglected but Uniquely Rich and Revolutionary Muslim Philosophy
As a theist muslim, gifted to be born and living in Islamic Republic Iran, I have had the opportunity to study, along with the works of most western philosophers, the eastern Islamic philosophical schools and traditions, and have been deeply fascinated with the superb contributions of Muslim thinkers, philosophers, poets and mystics who have provided humanity with a colorful and rich treasure of wisdom that has sadly gone similarly unheeded by mainstream western academia.
Open to all human intellectual heritage, I have also found a lot of brilliant wisdom in Eastern philosophies such as Taoism and Buddhism, and while central/eastern Asian philosophies are descriptive rather than analytic, I can still 'see' many truthful pearls of wisdom in them that offer close parallels to my established religious and philosophical beliefs.
As for the methodological distinction you made which is a widely held assumption among modern mainstream philosophical thought, it is to some extent true that Western Thought throughout its modern history has been mainly focused on structural and analytic thinking (that itself being a fundamental imperfection as I explained on the qualiquantitative vs qualitative here) in contrast to many Eastern philosophies that adopt a descriptive method of discussion and rely mainly on intuitive inquiry to truth.
However while there's been little attention or practice of the intuitive method in modern mainstream thinking, but both traditions have been dearly regarded and applied by muslim thinkers throughout the history of muslim thinking.
Al-Farabi (10th century) for example was the first and very seminal muslim thinker and polymath who adopted and expanded on the Greek philosophies. He built upon the Neoplatonic tradition which reconciled reasoned thinking with mystical experience by logically substantiating mystical statements. Al-Farabi working following this tradition managed to provide a philosophical explanation for the phenomena of Divine Revelation via Gabriel in his theory of human intellect. (Read this as a good exposition).
Generally, in contrast to the Western experience, where religious, intuitive and logical thought traditions increasingly diverged and their respective proponents became increasingly alienated from one another, all along the history of muslim thought, there has been a distinct current of muslim theist thinkers who did not dismissed any thinking traditions in favor of the other.
Instead of dismissing all religious wisdom as unverifiable dogma, they considered statements of religious revelation as sources of intellectual inspiration and subjects of deep philosophical contemplation.
Mulla Sadra (17th century AD) the last most prominent muslim philosopher represents the apex of muslim evolutionary thinking. As a poet, mystic, theologian and meta-physician with a fervent faith in Islam, well versed in all thought traditions of his past and contemporary history, he succeeded in fully achieving what Suhrewardi (12th century, the founder of muslim tradition of Illuminationism) had only attempted: synthesizing all valid theories and truthful wisdoms from past philosophical, mystical and religious sources into a consistent philosophical school which he called "Transcendent Philosophy."
Mulla Sadra was a strict observant muslim with a close affinity with intuitive experience and at the same time was a strong adherent to rigorous logical thinking. By establishing his Transcendent Philosophy, He practically demonstrated that religious faith, intuitive insight and philosophically inquiry are not dichotomous or disjunctive, that they go hand in hand and complement and nourish each other until finding their proper place in a lofty holistic philosophy.
Among the topics and problems he addressed and effectively solved in his philosophy are an ingenious logical proof for the Sufi descriptive belief in Unity of Existence, proving motion as a universal inherent change in the material plane of existence and its the fourth dimension (the philosophical equivalent of what Einstein proved via natural science centuries later), proving the incorporeal nature of human imagination (whereas muslim philosophers had only considered human intellect to be incorporeal until then) enabling him to philosophically explain the nature of Barzakh in Islamic Eschatology, and other stages of human posthumous life (a scientific exposition can be read here); the nature and different levels of beauty and pleasures therefrom (sensual, imaginal and spiritual) etc.
Some Sources and Resources
Therefore if you are interested in philosophies that unite reason and intuition you may want to study muslim philosophy in particular, for which purpose I recommend you the works of Henry Corbin, perhaps the first Western philosopher who for the first time "opened the eyes of the West, about the existence of a completely unknown world: the deep spirituality of the great mystics and Shiite philosophy developed in East Muslim world, particularly in Iran, after the death Averroes." or among living authorities on Islamic Metaphysics and Mysticism, William Chittick and Seyyed Hussein Nasr who currently lecture at American academies.
As for the first introductory read on muslim philosophy, you may want to start by Philosophical Instructions by Ayatollah Misbah Yazdi, who has written one of the most erudite and comprehensive text books on the topic ever translated to English language.
Here is also the website for Sadra Foundation in Iran, an institute dedicated to exploration and introduction of the personality and thoughts of Mulla Sadra, this unique genius of philosophy, mysticism and spiritual enlightenment in history.