From what I can tell, android phone/tablet YouTube app on an android device with a google account simply just doesn't let one sign-out anymore (unless one is fine to delete one"s google account from the device), not even via various ways that seemed to be possible only a few app versions ago. Pretty sure Gemini is lying, and nothing at all is possible without being signed in anymore. And, Incognito Mode isn't the same as not being signed in at all (in my opinion).
Previously, I avoided using chrome browser because it did something similar (would always automatically sign in with one of the device's google accounts, if one went to google.com).
Maybe I'll add a scratch google account to my device and then "switch accounts" to it in the YouTube app, to then see if the "remove account" from within the YouTube app account settings (remove the account from also the entire device) will result in the app being signed into nothing.