By the
Venture Theorem proved by blueblimp a long time ago (in...the same galaxy as here), this doesn't affect the probability of drawing a better card, as long as you don't know the order of your deck or trigger a reshuffle.
I would guess that this is equivalent to "+1 Card, +1 Action, discard some random number of cards from the top of your deck". If you don't trigger a reshuffle, it's exactly the same by the theorem above. If you do trigger a reshuffle, my conjecture is that your deck quality is the same as if you switched the location of your draw and discard. So, in mid game it improves your deck since you've bought more actions, and in late game it weakens your deck since you've bought more Victory cards.
Honestly, I can see this being a decently good $2, just for the cycling, but in most scenarios it's a slightly more exciting Pearl Diver. If your deck is very trashed out, it's $2 non-terminal Chancellor, but in that scenario you might be drawing your deck already. Could be very funny with Tunnel though, since it'll explicitly discard it unlike the other cards in the Chancellor family.