It's simple: all cards (except for Native Village which shouldn't be changed) would become stronger.
Wine Merchant and Miser would potentially become too good in the presence of other of these changed cards.
I don't think this is true. Some of the cards in question would become stronger in combination, but I don't see anything broken. Why do you think Native Village shouldn't be changed?
Having a chance to put reserve cards on the mat with Native Village is definitely not overpowered, it makes it the equivalent of +1 card, +3 actions at the very best. If you mat Duplicate or Distant Lands, that is. But having those cards on the mat heavily disincentivizes you from ever picking up. Similarily, Island - Native Village would clearly antisynergize. I can see how matting cards for Miser or Pirate Ship could be strong, but then again, picking up would make those cards completely useless, so you're stuck with a Necropolis for most of the game
Picking up any number of Wine Merchants is worth an additional +2 $ at best, and storing them on the mat is a lot worse than storing Bridges because there is less megaturn potential.
Haven, Gear and Save wouldn't behave any different unless there were other cards in the Kingdom that interacted with the Tavern mat.
After thinking about that,
if there are other tavern mat cards in the kingdom, these could easily cause problems. Namely, their interaction with Native Village is quite strong. You could manipulate your tavern mat almost at will, picking up the stronger cards you matted with Native Village and matting all the junk. You can already do that if there are topdeck interaction cards, but you have to be careful and get somewhat lucky to connect those. With Haven etc. it's just too easy.
Buffing Pirate Ship such that it quasi-plays the treasures you picked up from the trash (or, simpler: if an opponent trashes a Silver/Gold, put 2/3 Coin tokens on your Pirate Ship mat) is a good idea independent of "combine all mats into the Tavern mat" idea.
This is actually not what I intended. Playing the treasures from the mat would be highly confusing. Giving Tokens based on the kind of card(s) you trashed is wordy and wouldn't really work with kingdom treasures. As far as tweaks go, I prefer:
Original Pirate Ship:Choose one: +1 $ per Coin token on your Pirate Ship mat; or each other player reveals the top 2 cards of their deck, trashes one of those Treasures that you choose, and discards the rest, and then if anyone trashed a Treasure you add a Coin token to your Pirate Ship mat.
My version:Choose one: +1 $ per Treasure on your Tavern mat; or each other player reveals the top 2 cards of their deck, trashes one of those Treasures that you choose, and discards the rest, then you may gain a Treasure from the trash onto your Tavern mat.
This is still a stronger version, since, as long as there are treasures in the trash, you build the payload even if the attack whiffs.
During development of Adventures, there was a card that let you put any card you gained on your Tavern mat. I think it was a Reserve that you called when you gained a card, to put that card on the mat. It was cute with other Reserve cards, but mostly you'd just use it to put all your Victory cards on the mat. That didn't end up being good times.
Ya, that's pretty much why the Haven variant thing might be too strong. Other than that, I didn't suggest something like that, so I'm not sure where you're coming from. Native Village is still random most of the time.