Is it allowed to do a binary search with an oracle when proving NP-completeness?
Well, nowhere does the answer claim the reduction "implies that the original problem is in NP".  So, that explains your confusion.  You read something into the answer that isn't actually there.

Also, the answer says "Cook reduction".  This was a heads-up about the fact that it's not the Karp-style reduction you might be used to.  You might like to learn about the difference between a Cook reduction vs a Karp reduction.  Read https://en.wikipedia.org/wiki/Turing_reduction.  As you suspected, a Karp reduction is part of what is used in the definition of NP-completeness, but this answer isn't trying to prove NP-completeness, so the Cook reduction is "allowed".