It's not a failure of mathematics. It's a failure of mathematicians. There's no reason to think it's insoluble. They just haven't.

Mathematics is incomplete, but there's no reason to think that these specific problems are insoluble. A proof that they were insoluble would be remarkable, but extremely unlikely.

It is also unlikely that e + π is rational, and nobody expects it to be. Nor would it matter if it it were. The proof itself is more interesting than the fact of the matter, one way or the other. The mechanism of the proof is a tool for mathematicians, to go on and prove other things. It's a human problem, not one about the foundations of logic.