Late reply, but for me personally, I started doing it because my Keepass database is already accessed using two factors (password and key file). Therefore, I’d gain very little by keeping the second factor of those sites external - essentially, those second factors are compounded into the second factor for the database.
Cookies required for the website to work (like that one) are totally fine and, in fact, they don’t even have to ask you about them - if they’re not used for tracking. So no, asking each time is definitely avoidable.