Like yes, we have a really hard time talking about just about anything as finite object with physical extent, but jokes about frictionless spherical cows moving in simple harmonic motion started in secondary school. The gaps and shortcomings should not come as a surprise. But most of us also hold devices in our pockets that leverage actual quantum phenomena to function at all (diodes of any stripe only work because of quantum transitions). So while its true that there are a variety of unsolved and potentially unsolvable problems in physics, its a gross misunderstanding to say that it can barely answer simple questions.
I think about the Born-Oppenheimer approximation a lot, as its so obviously a hack to even do the math at all, but it undergirds basically all of solid state physics.
One simple example is what happens when you don't consider these as points but instead spheres. Also what happens when the spheres come close? The math starts breaking down, you start seeing infinities. I.e, in reality spheres come close and gravity doesn't go infinity.
Besides, very often the simplified case gets you surprisingly far because the difference between idealized situations and reality is often negligible or at least easily describable - see perturbation theory. The simplified cases are well worth studying.
I guess I didn't do that much physics, because for me it comes up more in other fields. In statistics, for example, it is critically important to understand the limitations of your results. For example, you might assume that error is normally distributed. You don't want to forget about that assumption, because it is very commonly violated, and it can make a large difference in your conclusions. Yet in school, it was almost always handwaved aside with "Law of Large Numbers mumble mumble mumble". Even when the law didn't apply, or the definition of "Large" happened to be "way bigger than your pathetic number of data points".
It's also why there's often such a gulf between academia and industry. Academic results walk a tightrope of assumptions and preconditions, and trying to put them into practice always finds places where those don't hold. Sometimes they even start out holding, but then everybody takes advantage of it until competition drives everyone into optimizing the residuals. If there's a space where things make sense, competition will always drive you to the edge of that space. Or beyond; competitive pressure does not care about keeping your equations simple and pure. Back to the point, you might study a field for years and then land a job in exactly that field, only to discover that everybody is looking at it completely differently because they've exhausted the simplified space and are deep in the land of heuristics, guesswork, and approximation. The market for spherical steaks was saturated years before.
(I have my own beef with the "sweeping under the rug" which happens with (electromagnetic) pseudovectors, but I do realize that requires a LOT of effort to fix.)