Mysticism, the realm of non- or anti-conceptualism, can certainly be a domain for rascals, to make statements that are so contentless they sound deep, and thus avoid further discussion or critique.
It should be noted, Zen masters were not in such a position. Many times students or wise women would ask a question of a master in front of an audience, which while seeming relatively straightforward is actually a challenge about subtleties of doctrine, or practice, or realisation, or spiritual capacities. Most koans occured originally in the 'transmission of the lamp' genre of Ch'an literature, that is they were about the transmission of awakening that confirmed masters as holders of the lineage. So there is a lot more going on than simply running up against the limits of language. An example.
This is the story of Yunyen's (or more often attributed to his student Dongshan Liangjie, founder of what became the Soto school) final enlightenment: http://www.abuddhistlibrary.com/Buddhism/C%20-%20Zen/Modern%20Teachers/John%20Daido%20Loori%20-%20Dharma%20Talks/Discourse%20Yunyen%27s%20Insentient%20Beings%20Preach%20the%20Dharma.htm
The context for this is the arising of the doctrine of the Buddha nature of insentient things http://buddhism.org/kr/koan/Robert_Sharf-e.htm
more or less, the idea of non-dualism between minds and world. This is problematic for wider Buddhism, which interprets phenomena in the world as the results of mind, as arising out of karma, making for an apparent contradiction between early Buddhist texts and the Nirvana Sutra. Yunyen's understanding and response, is not just about the fundamental emptiness of conceptual categories, it is also about dealing with exegesis and practice, older and newer terminology, and the ongoing process of affirming that awakening is universal and accessible by all, not only those who are monks and scholars (the view which gave rise to the whole 'sudden illumination' approach). The wording is careful, and precise.
I would observe from this, that anti-conceptual thinking is not necessarily anti-intellectual thinking. There is a big picture, the aims of Buddhism to end suffering, and provide tools for people to let go of unhealthy ways of being. Buddhist thought has to be understood in that context, metaphysical speculation is endlessly condemned as a pursuit for it's own sake. But, it can serve a healthy purpose, where set up as the ladder which is left behind after climbing it, the boat which is left on the bank after crossing the river. Ontology and epistemology are affirmed as not for their own sake, but being aimed at finding a healthier more wholesome way to be. See Seung Sahn's Zen Circle teaching for an affirmation of how manifesting oneness, emptiness, magical thinking, and being directly present, are not 'fundamental truths' but only tools to affirm our fundamental freedom with, and be harnessed only 'to save all people' https://www.kvanumzen.hu/en/weekly-teaching/zen-circle
I interpret Wittgenstein's core insight as that our ability to make sense of the world, is not located 'privately' in ourselves, in a Cartesian pineal gland, but in our community of understanding, most clearly expressed in language. Expressing the non-localised nature of our consciousnesses. This can be related to karma, where also we are subjective manifestations of wider conditions, phenotypes of causative genotypes, able to act to some small degree on these structures but absolutely relying on inheriting them to exist at all. In this picture, we have to understand emergence of cultures of thought on language, just as we do impacts of our biological inheritence on language, we cannot attain 'direct access' to the world through language. There are also in this view language games which don't neccessarily need to be refuted, but can simply cease to be played, judged by a 'therapeutic stance' outside of language. This is Wittgenstein's motivation and purpose, certainly in his later work.
The 'rascal's' non-dualism and anti-conceptualisation, must be distinguished by their motives. Is the aim to end discussion, evade critique, gain control or power? Or is the recognition of these aspects of reality being used to untangle contradictions, to cut Gordian knots (the bodhisatva of wisdom has a 'diamond sword' for this), to liberate and empower people? Wisdom must be useful, or be able to be made useful. Conceptual linguistic words are not funame tally bad but can be, 'beyond words' is not fundamentally better, it can also be misused. Our fundamental motivations are a choice, arising from a fundamental freedom in how we choose to redefine ourselves, and reorientate all other ideas. In that sense, they are beyond words. But like a ladder, or a boat, words may be esse tial for us to navigate to that freedom.
There is probably a lot more that can be said, on Kant & Hume's different motives, but enough words :)