Why do we care about black hole interiors' physics?

Whatever happens in there is not falsifiable nor provable to the outside.

General relativity predicts this, but a) no one has ever checked experimentally, and b) it seems to be incompatible with the rules of quantum physics. Every attempt at mixing quantum theory and GR has produced results like Hawking radiation that tell us that the black hole interior and exterior are not completely decoupled.

Simply because our final goal is a set of laws of physics that describes any part of the universe equally well. Let's say a physicist jumped into a black hole and saw that the interior of the black hole was composed entirely of John Lennon clones. His last thoughts before getting spaghettified would be "why?". From his perspective, physics is incomplete.

Sure, we probably can't use it to predict anything -- but modern physics is much less about predictions and much more about having a beautiful, mathematically rigorous model of the universe. Mathematical models with discontinuities usually aren't "beautiful", and John Lennon in black holes counts as a discontinuity if we take general relativity as our mathematical model of the universe. Which would mean that we will eventually have to replace our model (which is why knowing as much as we can about the inside is important). Besides, if our current theories partially fail inside a black hole, we need to patch that up.

Why do we care about the physics inside black holes?

By Karl Popper's reckoning and the rules of general relativity, we probably shouldn't. From the outside, it's not being scientific to theorize about the inside. Whatever happens there is not falsifiable; not just difficult to falsify (like physics at the Planck scale or theories about the inside of the sun), but impossible in principle.

It may be mathematically useful for other reasons or philosophically interesting but it's not, according to Popper, scientific. If suicidal people jump in and survive, fine, it's valid science for them (while they last) but for us on the outside, it will never be scientific to theorize about the region.

That should be sufficient reason not to care, but there's something even more fundamental. Beyond the falsifiability issue, the greater reason not to care about the inside of a black hole is the more and more likely scenario that there isn't any inside to care about.

Raphael Bousso muses that space and time seem to "somehow" end at the horizon. And Joesph Polchinski sums it up this way: "... the inside of a black hole — it may not be there" ... "Probably that's the end of space itself; there's no inside at all."

Granted, these new ideas are speculative and controversial but what, really, is the evidence that the spacetime manifold does continue across the horizon? That looks to me like an unexamined assumption. The idea that an in-falling observer simply drifts on through the horizon assumes continuity of the manifold. The same assumption is there for the theory that entanglement bridges the horizon.

GR tells us that in-falling matter and energy never reach the horizon; firewall theory puts a vaporizing surface there; the membrane paradigm and stretched horizon theories completely ignore the inside and describe a one-sided surface with physical attributes; and Lynden-Bell and Katz's calculations put the mass of a black hole entirely in its external gravitational field.

These ideas strongly hint that there is no interior to worry about.

Unfortunately, all of this will upset a lot of very smart people who have invested so much time in theories about the interior, and rightfully so. But it also presents a great opportunity. If black holes are actual holes or cavities in the spacetime manifold, it ushers in an entirely new and untapped paradigm to explore. We live in interesting times, indeed.