The consent form is a third-party plug-in which needs to set a cookie. If you allow third-party cookies on the site it’ll work, if you block them it will fail
I have third-party cookies blocked (no exceptions) on all platforms and browsers and I don’t get the popups.
The popup only needs to set a cookie because it’s badly written: it should try to write a cookie (which will of course be a third party cookie), if it can’t then the browser user doesn’t want third party cookies by default that includes all the tracking and advertising cookies no need to do anything more like pestering the user. If it can write a cookie then it asks for preferences.
Logging out? The standard time periods for WordPress login: 2 weeks if I check the “Remember me” box, two days if I don’t. I’ve posted the fix to change this to indefinite (or any other period) before so it’s not fixed.