This is pretty brilliant. My biggest concern is that if my password for a site gets compromised, it gets a lot more complicated. Presumably I'd have to memorize a separate master password for retrieving the new password for that site.
This blog post[0] shows how LessPass lets you change passwords per site, and get around tricky password requirements.
[0] https://blog.lesspass.com/lesspass-how-it-works-dde742dd18a4
The profile is also used to store other site-specific attributes, like if it has some weird password rule that requires pure alphanumeric or a specific password length.
There is still the benefit that the passwords can't be exposed by a compromise of the service. Presumably it doesn't store your master password anywhere, so all they'd get is the site and counter info.