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.
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.
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.
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