Philosophy, Mathematics, and Science
The Ancient Greek geometers did not practice geometry with measurement which might be taught in an elementary school today. The straight edge and compass were their tools, and the practice was highly philosophical sometimes mystical. Pythagoras and his followers are said to have had mystical and supernatural beliefs and practices, for instance. What the ancient Greeks did not do in their geometry was brought to life by Descartes and his Catesian geometry which was a leap forward in method as it wedded two traditions. These days, while it is pedagogically interesting to practice geometry construction, one finds no major math conferences devoted to it. As a method, it is dated, and the field has evolved. It's still fun to bisect an angle with a compass, but serious math is done with tools like Mathematica which does not only constructions, but differential equations and neural networks. Philosophy, like it's closely related twin children, math and science, too has evolved, and to be on the cutting edge of philosophy means one has to pay heed to new facts and truths.
Metaphysical Traditions and Complexity of Philosophies
It is right that you note a rough bifurcation of metaphysical practice, with Metaphysics and grand and sweeping theories characterizing one methodology, and metaphysics whose philosophers actively work to purge it of content. Wikipedia has a subsection devoted to the rejection of Metaphysics in this sense. It should be noted, however, that philosophies do not distill neatly into black and white categories; it is a pseudoproblem to characterize discourse as something that occurs between a group informed by mathematics that abandons metaphysics and a traditional group that lies on rhetoric and natural language as two groups that "talk past each other"; philosophical positions are far too complicated to accomodate this view. In fact, Western philosophical traditions include Thomists, Marxists, existentialists, phenomenologists, and analytical traditions, and those are broad umbrellas which struggle with subdivisions and demaraction issues. They are all philosophical in that they attack ontic, semantic, and epistemic questions (to name a few), but they can divide and overlap in myriad ways. Strongest still, many philosophers have differnces in their local theories regarding different phenomena, so that one may be both an idealist and realist whether one is discussing mathematics or science. I suspect it may be just as hard to put philosophers in buckets as it is to get them to agree. I'd be interested in the defense of your proposition analytic philosophy is a form of idealism. Most analytical philosphers I've come across seem to be realists to varying degrees, with the whimsical Daniel Dennett representing the eliminative materialist camp. To claim Dennett is an idealist would be nonsensical (in the non-pejorative sense).
Diversity of Thinkers
While I think questions regarding encouraging types of discourse regarding this forum are more of a meta topic, clearly one must admit that there are a range of abilities and interests represented here, that no simple dichotomy is adequate. Dilettantes such as you and I are clearly representative of our community who have more than a inchoate and passing interest, but hardly can lay claim to any expertise, and we rub elbows with those who seek to have their homework done and those who clearly have a preternatural command of the Canon; and interests vary widely. I believe there is regularly convergence, and it is manifest in every contribution which receives an upvote, downvote, comment, or rebuttal.
Analytical and "Traditional" Views
If one wants to explore the gulf between the analytical-phenomenological approaches AND other philosophies because of the former's reliance on the philosophy of mind, language, math, science, and logic along with and the tendency to avoid Grand Sweeping Theories about Everything, that's a valid question whose answer might parallel Gould's doctrine of NOMA. As some philosophers have recognized, there is a social division of labor to thinking, and philosophical inquiry is just too broad and demanding for person to understand everything and make claims about all possible worlds and all philosophical objects. Language, math, and the mind, for instance, all have metaphysical implications, and even wrestling with a small part of the inferences can take a lifetime. Amateur philosophers know everything; professional philosophers specialize. The consensus seems to be no one book or philosopher can speak definitively to philsophical truth, and that speculative metaphysics is best understood as one's private religion. This intellectual separation is intellectually productive in the same way the separation of church and state is.
On the Importance of Psychology and Mathematics
The history of Western philosophy (there is a world beyond the West, after all) is a narrative which starts approximately with Thales of Miletus who along with the Presocratics, infused logic (in the broad sense) and naturalism into culture. No longer did one need rely on divine revelation and religious mythology for truths nor did one require gods to understand the world. As Christianity displaced paganism in Europe and then embraced philosophy in the form of Scholasticism, the Renaissance birthed natural science from natural philosophy. But natural science evolved (that science is somehow a monolithic static entity obsessed with objectivity is severely deficient), and in the 19th century, modern science blossomed and is still evolving. Besides the introduction of evolution, psychology, mathematics, and formal logic came into their own. _Great thinkers like Frege, Husserl, Wittgenstein, Russell, and Quine have since tackled questions that go beyond the simple contemplations of doxa and episteme. They took took epistemic, ontic, and semantic questions in new and exciting directions. From Husserl and the Sciences, p. 90:
Why should we bind the analysis of these concepts to the metaphors and periphrases of an explanatory commentary? ... A straightforward answer... consists in subordinating all philosophical questions to logic [as Frege advocated].
Frege and Husserl and Quine all have advanced the theory of knoweldge by examining the interplay between subjective and objective mind, logic, and mathematics since (introspective and naturalistic) psychology, mathematics, and logic all have metaphysical implications, in the same way Mendelian genetics and Darwinian natural selection do. Certainly great philosophers who came before such as Aristotle, Hume, Spinoza, and Kant have a place in history, but as their knowedge became dated, philosophy evolved. Of course, many philosophers do find their idols and demur.
Every philosopher struggles with ideas; it's the nature of the beast. From naive realism to eliminative materialism to subjective idealism, philosophical positions vary, and to the extent that philosophers live in a real world so too does their philosophy. I routinely see exchanges between thinkers here on SE Philosophy, and myself have evolved on many issues not limited to seeing causation at play on the supervenience of mind on matter to adopting a position closer to nihilistic ontology. There may be plenty of talking past each other here, but the critical exchanges far outnumber those experiences.