Rainbow tables are hash-realpassword pairs and can be very fast in telling you the password given the hashed-password

Yes - I get that salting helps from attack using a rainbow table.

But passwords are typically rather short so not too difficult to crack by brute force. Assuming the password is made up only of upper case, lower case letters and numbers then there are only (!) 218,340,105,584,896 permutations. That is 62⁸. Whilst that's a lots of permutations, I read recently that 2011 technology could run through all those in 23 minutes. Imagine what 2021 technology can do and there is no need to go through them all. You stop when you get it right!

So, to take a simple example - let's say the hashed password was 1234abcdxxyyzz. Without salting you assume a minimum length of 3 characters so start at aaa then aab, aac, etc. If instead you have salts stored in the same table as the hashed password you just add the salt to the end and try the permutations. So instead of aaa you try aaaSaLT123, aabSaLT123, etc. until you get the result 1234abcdxxyyzz.

Or am I missing something? Is that not how it works?