If that was the case, that elaborate listing of all things that might look like a proof to a naive student, but are actually not proofs (and not even just partial results, but fundamental misunderstandings of what constitutes a proof) could have been easily and equivalently replaced by 'I am interested only in a full proof, don't bother me with partial results'. Yet, they were not.

To a real mathematician you would not have to list those explicitly, he/she would have understood that implicitly from 'give me a full proof'. That listing makes sense to say only to somebody who pretends to be a mathematician, but has not true understanding of how the math works. The models are getting better and better in this pretension, but prompts like that reveal that it is still just a pretension, not a true understanding.