Yup. The main purpose of salt (aside from increasing entropy by increasing the length and complexity of the hashed value) is that it prevents rainbow table attacks where an attacker pre-computes (or downloads precomputed) hashes for common passwords, dictionary words, and brute-force style variations of same.

A hash unique to the site would require the attacker to create a site-specific rainbow table, but once created it can be used for all passwords. Having a unique hash per password means that the attacker would have to generate unique password tables per user, which for a suitable salt & algo is impractical, even if (as they normally are) the salts are stored with the passwords.