Is there a C->JVM compiler out there that's any good and actively maintained? What are the major problems that come up with this approach?

I don't think there's anything that's fundamentally impossible (or even conceptually very hard) to solve, it's just that most of the times you're solving the interop problem with JNI/JNA. The opposite route is massively more complicated to implement and nobody wants to dump their time into that. IIRC, as a matter of fact, there are many amateurish level C->JVM compilers but nothing that you'd be comfortable working with on a daily basis.

I guess it boils down to "yeah but why".

One possible approach today is to compile C to WASM, then run WASM on the JVM as GraalVM supports WASM. https://www.graalvm.org/reference-manual/wasm/

If you're using GraalVM already, I think Sulong might be better -- my experience with WASM (for C) is that there's a bevy of POSIX-correct-but-ANSI-incorrect tricks that don't work properly on WASM. Debugging is also much nicer with LLVM IR (worst-case, just compile it to a binary and gdb that) than WASM (Firefox doesn't show the stack, Chrome has sourcemap bugs).

NestedVM works pretty well. http://nestedvm.ibex.org/