We talked about password hashing and salting in class today. Our professor had a very different understanding of the use case of salts from mine and said that you might not store the salt at all and just check every login attempt with all possible salts and authorize if one matches.

I don't really see any point whatsoever in this since now my service is much more vulnerable to brute force attacks (send one password, server checks many), but I guess it is a little more secure to pure dictionary attacks against the hashed password.

So is there any use case where people would actually do this?

Not storing the salt is bad advice.

The main purpose of a salt is that each user password has to be attacked individually.

If you do not store the salt then, as you said, you need to try every single salt combination in order to validate the password. If you need to check every single salt combination, this means that the salt cannot be too long (you mentioned 12 bits in a comment). If the salt is not long, it means that the salt will be repeated for many users. If it's repeated for many users, it means the attacker will be able to attack many users at the same time, which will save the attacker time.

By doing that, you nearly completely defeat the purpose of using a salt.
