I am a humanities TA, I've never rejected an extension as long as I get asked before the due date. I'm pretty sure I gave someone extra time when they e-mailed me 2 hours before the due date. Half of the time, I barely read the reason why the extension is being requested.
As for this, if the strategy works it's probably because they would have given you the extension anyway if you just asked. If they are being draconian about it, this probably wouldn't work anyway. Depending on what you use to turn in the assignment, it gives you a preview before you hit the final submit and/or after you submit and the professors know this now. Meaning, it would be up to you to fix it or to send an e-mail immediately (for humanities work at least). Even if it bought time, you'd still be losing points for turning in the assignment late. Just ask for the extension, you will most likely get it.
As a final aside, please don't use AI for humanities papers. You don't write like that. We have to use the most trash, questionable "A.I. checkers" possible if something smells fishy. Professors are already paranoid of A.I written papers. Last year the checkers had to have a sky-high percentage assuredness to get you in trouble. That bar has gotten real low this year. Just turn in half-assed self-written garbage. The worst thing that absolute trash gets you is like a mid 50s or a 60s. Getting caught on A.I papers can make you auto-fail the class and get reported to community standards or whatever the equivalent is at your college.