I do the coding screening interview for my company, a semi-large-ish private Bay Area company. If you tried to pull this, it would just result in either you failing the screen due to not completing enough of the problem (or any, depending on if you were willing to do the screen question we asked you), or you'd waste the 10 mins we reserve at the end for you to ask any questions you have when you could be asking a fellow engineer questions about what the work is like etc

You wouldn't fail due to my discretion by the way, I don't have a lot of leeway - to reduce bias, I can only pick one of a few questions to give, and I have to grade you against a rubric. So to pass you'd literally need to get xyz things on the rubric, and if you spent much of the time trying to screen me, I wouldn't be able to check them off for you.

The rigid format is not how I'd do interviews personally? And if it were more free-form id probably be at least intrigued by what question you posed me. But interviewing isn't very well-rewarded work at my company so the dominating strategy as an IC is to do anything you can to minimize the time footprint that interviewing inflicts upon you. Alas.

Anyway, my point is, it's an intriguing move but you're gambling on what kind of interview process you're in.