... but will be using a randomly-generated password for every website that asks me for a password.
I'm not in the position to argue the merits of the rest of your post, but this last part seems obvious.
Once you have more than a half dozen passwords, your ability to remember them drastically decreased unless you are some sort of savant. You need a password manager. Once you are using a password manager, there is no reason NOT to use a different, random, difficult to hack password on every site. I have no idea what the vast majority of my passwords are - the only ones I remember are the three I use multiple times a day at work. The rest get created and filled out by 1Password.
And, yes, of course, now I'm at the mercy of my 1Password password and the company's ability to manage their program. Can't be perfect and the current system really does suck but this way seems to be the best of the worst.
One feature I wish Agile Bits would set up is the ability to automatically change passwords on a regular basis. As it is, I manually change some high value passwords every so often (not Slashdot's of course). PITA.