From what I understand, salting prevents computing the tables ahead of time and the ability to cross-reference a single password hash against all other password hashes. It doesn't prevent people who have access to both the salt and the hash from performing the same attack against individual passwords.
Yes, that sounds correct. I thought the idea, though, was to make the salt non-obvious for this reason. I would hope people aren't storing passwords in a table with a "salt" column, but I don't really know. But I think you're right: if you know the salts and you want to use this approach to go after a small number of high-value targets it could work.