The stay logged on and trust this system actions are probably controlled by a browser cookie, so won't work if you don't allow cookies from that site.

In my experience, they are. In my bowser, I have to explicitly allow the saving of site data if I hope to be able to keep myself logged in. For me, this is fine; I only want to stay logged in on a handful of sites anyway so the tradeoff between adding an exception for some sites vs. allowing every single site I visit to be able to save some piece of data doing god-knows-what (likely helping Facebook and the like track where it is I visit) on my machine is worth it.