Would this smart ring fit on pretty much all types of drain covers or is this size specific?

Since you have already verified that the main drain line is closed, it sounds like you need to install one of the smart ring drain covers below. They are made specifically to work with in ground pool cleaners to prevent them from becoming stuck on the main drain. They are easy to install and should eliminate the problem.

http://www.backyardcitypools.com/Automatic-Pool-Cleaners/NE290-Smart-Ring-Drain-Cover.htm