the basic cards don't have randomizers, do they? like, there's no Gold randomizer, so this can't prohibit gold. that means it only takes four senators in a four-player game to hand out all the kingdom card randomizers. unless I'm wrong about that, but I don't see that there would be a purpose in a gold randomizer since they are in every game.
wouldn't it be simpler to have it just say to set aside an "extra copy" of each card? you can use the randomizers to accomplish that anyway.
also this seems really bad in most two-player games, because unless we both go for it, if I'm going for a strategy that only uses, say, 3 kingdom cards, you have to play your senator 8 times before it does anything to me. if you use a version that includes the basic cards, that goes up to 10, since it's likely I can take curse and copper without hurting myself in the least. maybe if you can put together some +cards/+action engine with a couple of these and shut me off from multiple cards per turn, but that sounds really slow to set up. in a larger game it can get there faster, since more people are taking cards, but if your opponents are going for different strategies, they can all dodge it by taking each other's cards.
also, that introduces some politics, in that when you play Senator, and there's two left, neither of which affect me but one of which affects the guy downstream, I can choose whether I want to help him or screw him over. but I don't think it's worse than some existing options in that regard so I would not worry about that.