As for replacing terminal draw, well I think non-terminal draw is a much better comparison to make -- nobody is spending any Actions at any point to make this happen. I'm pretty sure the answer to this is no, though. You have to be consistently getting more value than $3 and a buy from two cards in your deck to make this even close to worth it in that case. This seems like something to do with those three dollar hands where you don't really want to buy anything.
It should be noted that Expedition draws cards immediately, and is therefore significantly less valuable in the presence of discard-to-X style attacks. Approximately 22% of kingdoms include at least one of: Minion, Ghost Ship, Militia, Goons, Margrave, or Urchin/Mercenary. An additional ~11% of kingdoms include at least one of: Tournament (Followers), Knights (Sir Michael), Peasant (Soldier), or Taxman.
An aside on procedure: If events were intended to be included in this alphabetical exploration of Adventures cards, the following were skipped: Alms, Ball, Bonfire, and Borrow.
I wonder how powerful of an effect opening with this on 3/4 would be. It would give you a high probability of spiking $5 or $6 on your second turn, but would also cause a shuffle. This mechanic alone could have a great effect on the entire game, depending on what the $5-cost cards are and if you want any <$5-cost cards. It seems like you'd be at an advantage on 3/4 opposed to 4/3.