No idea why this popped into my head, but just how broken would a mulligan rule like the following be?

"Each time you mulligan, shuffle the number of cards in your hand into your library, then draw that many cards minus two. When you keep your hand, draw cards until you have a number of cards in hand equal to your maximum hand size."

So, you quickly see less and less info, but you never start with less than 7 cards.

I feel like I'd mostly use it to fish for combo pieces or "need to have" cards and then, only I know I have what I need, I'll be happy with whatever else pops in my hand.