I don't understand why the "old way" doesn't work. Most website that ask for a password aren't getting any dangerous information about me, and I use asdf1234 or similar throwaway passwords on all these sites. Including this one! (no disrespect intended).
A lot of it is businesses asking for you to set up an account as a condition of them sending you their "whitepaper" (brochure).
Or are you suggesting that there is enough information that's being saved even on these websites ?
I too am curious about the password managers, but wonder if they are themselves vulnerable. Maybe a bad idea keeping all your passwords in one place ?