Most of the topics we discuss aren't part of the science curriculum for even Astronomy and Earth Science doctoral students. They don't like teaching about the problems of the model.

No. For instance, "dark matter", "dark energy", and alternatives to explain unexpected rotation rates of galaxies and acceleration in the rate of expansion of the universe are very active and well-known topics of research in physics and astronomy today. Like the earlier issue of the hitherto unexplained precession of Mercury's orbit that was solved when the relativistic nature of gravity was discovered, these "problems with the model" could lead to breakthroughs in our understanding of the universe.

What "problems with the model" do you think are given short shrift in geosciences that could be better explained by earth being flat instead of spheroidal?

For example, except for a niche class of astrophysics researchers, few PhDs barely know what the Three Body Problem even is.

No, again. Solving two-body orbits is a staple of undergraduate classical mechanics courses. The difficulties of the three-body problem were introduced at the same time when I was an undergrad (decades ago), with the claim by my professor that anyone who could find a complete solution to the three-body problem would become famous. More recently, whether it's impossible or not, the search for a complete solution is less urgent because the availability of inexpensive computing power has made accurate numerical solutions to the N-body problem much more practical than back then. It's still well-known amongst physics students, and I suspect every candidate for an degree in astronomy (and physics) would have taken classical mechanics before graduating.

When students do find and show an interest in it, they are often discouraged from looking into it as a thesis topic and are told that it is an impossible problem that will hurt their career to be associated with or to try to contribute to. They are also told the same when they show an interest in problems with Relativity.

At this point, any search for a solution to the N-body problem (for N >= 3) would more likely be appropriate for realm of mathematics than physics (and astronomy). If it has been proven mathematically to be impossible, then a new type of math would be necessary (kind of like the development of Calculus), and would be an inappropriate subject for a physics dissertation (What would the title be? "Another Failed effort to Solve the 3-Body Problem").

Why do you think the lack of an analytical solution for this is "a problem with the model" anyway?