Earlier quoted context omitted.
Further, in certain cases (P =? NP is one I know of) it has been proven that certain types [0] of proofs cannot possibly work -- and yet people keep offering "proofs" of those bound-to-be-wrong types. [0] Not sure if 'types' is the right word? Using it colloquially. Maybe 'classes' would be more accurate?
It's subtle, because some of the impossibility proofs are effectively saying that any real proof needs to be somehow sensitive to whether it's in an imaginary universe with oracles, so they can _fail_ in them!
It really is mind-bending stuff -- I have only the vaguest idea of how these proofs could work, but you summarized it beautifully.
As an aside, this is also perhaps also one of the best motivating examples of the idea of "oracles" in math. ("Ok, assume we can do $IMPOSSIBLE_THING... now what?" :) )