Now, lets say I've got both a main computer, and I use firefox and chrome. And then, I've got an iphone, that only knows safari. And I need to use a chromebook at times.
How do I manage to synchronzie all of these, so that my unrememberable, non-written down, computer trusted passwords are usable on these other machines?
The real solution is some sort of physical key ring, a USB-thingie that, just like physical house keys, unlocks accounts. Now figure out how to make that safe when used on someone else's untrusted computer.
In that case your password library would have to be on a remote server in order not to pass the data through an unsecure computer. As for how to safely log into your password manager, you would need some one-use codes which require you to have some physical object generating those for you. For example in banking a smartphone is often used for authentication exactly in this fashion with the only change in that the password is generated remotely and sent to your phone.
In a war it doesn't matter who's right, only who's left.