More generally, what if I participate in a pool and just keep the good hashes for myself? I guess this would be strictly more profitable for me than just participating in a pool. But if every one does the same, pools should just never find a block. Right? Then: Why do pools exist in the first place?
You can't "keep the good hashes". Either you send them to the pool, or you throw them away. If you throw them away, you get nothing. Yes, you can do that, but it's a waste for you.