I think you are imagining a different class of "questions".

To clarify, I was doing research on applied math. My field is not analysis, but I needed to prove some bounds on certain messed up expressions (involving special functions, etc), and analyze an ODE that's not analytically solvable. I used the COT model a fair bit.

I would ask ChatGPT for hints/ideas/direction in proving various bounds, asking it for theorems or similar results in literature. This is exactly the kind of thing where a researcher would go "yeah this looks like X" or "I think I saw something like this in (book/article name)", or just know a method; or alternatively say they have no clue. ChatGPT most often will confidently give me a "solution", being right 10% of the time (when there's a pretty standard way to do it that I didn't see/know).

On the whole it was quite useful.

It's pretty easy to test when it makes coding mistakes as well. It's also really good at "Hey that didn't work, here's my error message."