Would this be covered under the tos with the API?

I'm thinking about making a game rating site similar to flickchart, and one of the issues I'd end up having is building out the database.

Now, while I could manually compile a lot of info, it'd be nice if users of the site could search for games in the site's db, and if it's not found, have it search on here and pull up the closest matches. From that point users could pick the match, and the few fields we'd need would be copied to our db, as well as the id from GB, with a link to the games page on the site.

This way the site wouldn't constantly be pulling from GB or outright taking the whole db, and also it'd be promoting GB through the links back.

Would this be allowed? If so, would that change if the site was ad supported? Or supported by patreon or donations?

According to the TOS, you can not make money, with ads or otherwise, from a site using the GB API in the way you described.