> If you use the same salt for all passwords then I can easily create a rainbow table consisting of "keyword" + salt hashes and doing so I can crack multiple passwords.

That's not a rainbow table, a rainbow table is a list of precomputed hashes (which you'd just do once on a cluster you pay — of buy from a guy who did — and then store/stash)

But you're brute-forcing the whole table (you can look for matches after each computation) instead of having to individually brute-force each password. But you wouldn't keep the forced salted hashes around since there's no need for them once you've tested the leaked hashes.