Should Counter Buffs Be Cleansers?

Buff immunity makes you immune to gaining buffs, not to ones you already have, so no it shouldn't remove anything. CLS has a special attack that says it dispels, and adds buff immunity. Those are two separate functions, they are meant to be such, and I don't see a need to alter that mechanic.

As for the others, no adding a buff/debuff shouldn't dispel/cleanse its opposite. It already negates it while it's there. If you have health up for two turns, then acquiring health down for one turn effectively negates it for one turn, then when health down wears off you get the benefits of your buff for one turn. It makes the length of buffs/debuffs matter.

There is nothing wrong with what you're suggesting, but I personally don't see a need for it. There isn't anything wrong with how those mechanics function already. There could be some unintended consequences of monkeying with something that isn't broken to fix a non-problem.