When framed correctly, a stateless browser is totally acceptable for general users. General users have no concept of cookies, so if you describe browsers as just a dumb window to let you load websites, it totally makes sense that you would have to log back in when you close the window.
For example, I set my Mom up with a Firefox configuration that doesn't remember anything when you close the window (basically incognito by default). She has used that configuration for years and never complained. In fact, she always compliments me for how safe and reliable I've made her browser. When she sees other people on other computers just load their logged-in websites without having to log in, she thinks they're totally unsafe and exposed.
She associates closing the window == back to safety, so whenever she ends up on a scary website or sees a scary popup, she just closes firefox and opens it back up again. It's wonderful. I didn't have to teach her about cookies or sessions or anything. All I had to teach her was that if she ever got into a situation where she didn't know what to do, just close the window and you're safe again.
Anyway, I feel pretty strongly that if browsers were incognito by default and you had to opt-in to persist sessions (e.g. whitelist cookies), general users would get used to it pretty quickly and end up thinking the time back before was a very unsafe place (like the free love era before the AIDS epidemic).