Public key encryption transforms are one of the biggest foot-guns in cryptography. OAEP is at least not prima facie broken, but using it still exposes you to the the design risks of building with public key encryption.
(There are attacks against OAEP, but they're less common and not intrinsic to the design the way PKCS1v15's are).
Even if RSA does go away, and it will, public-key encryption primitives won't. The post-quantum craze is mostly made up of encryption and signature primitives, so we have that to look forward to. Even the lattice-based key agreement we have is basically fancy KEM, which leads to things like [1].
My impression is that RSA never really got the "djb treatment". The people designing OAEP and friends were mostly theorists concerned with security reductions, not implementation issues. I think an idiot-proof RSA scheme could be devised, but it is now way too late for that.
(There are attacks against OAEP, but they're less common and not intrinsic to the design the way PKCS1v15's are).