KDF-based password manager
What you describe will work once, and it will be fine to use one salt, which need not even be secret as long as it's different from everyone else's salt.  Just derive each service password from the same master key with a different HKDF info parameter.

But what happens when a site's password database is breached, and turns out not to have been hashed, and you need to generate a new password for the site?  What happens if a site imposes restrictions on the character set, length, or structure of the password?

See Tony Arcieri, ‘4 fatal flaws in deterministic password managers’, blog post, 2016-11-22 for further discussion.  (The title should perhaps have been about stateless password managers rather than deterministic ones.)