I got lost in the reeds, I though we were talking about bonus action unarmed attacks, which, for doing a bonus action attack you need a light weapon and only monks consider unarmed attacks "light" for the purposes of bonus action attacks. So yeah, fighters RAW can't bonus action unarmed strike with their "offhand" because they don't consider their first/feet light weapons.

In any case, you are right, anyone should be able to use their whole action to kick someone even if they are wielding a 2hander, but only monks can do it as a bonus action effectively, if they are using versatile monk weapons like quarterstaves.

I know you wanted for references, but since I changed my argument you might not need them anymore. If you want I can search proper references.

They specifically don't require a free hand