I don't think either turns or points is a good way to do stars because the value and achievement on both is entirely board dependent, and creating a formula that takes into account all the cards and combos of cards to determine what constitutes a 2* or 3* performance would be nearly impossible.
Well the formula gives you an estimate, and then after a bunch of people have played it, you can tweak it based on how much they actually scored (I guess without changing scores for people who played already, since people wouldn't like losing stars this way). Currently I don't think they factor anything in, although I'm not sure.
Therefore what I would suggest would be a bespoke system of achievement for each individual level. On some levels it would be turn based, on others it would be points based, on others it could be winning while buying a certain number of a certain card. You could even have really quirky stuff for getting 3 stars on some levels like "win without buying a silver" on a predominantly BM board. While it will be a slight PITA going through all the levels and finding a suitable challenge, it would make the adventure mode a lot more fun and a lot more replayable as a result.
I was initially fond of my suggestion that two stars be "didn't buy silver/gold" and three be not buying some particular other card. Then I realized that I personally would just never buy the indicated card - why beat this level twice when I can beat it once? It wouldn't be creating a new challenge so much as it would be only playing with nine cards. And of course you'd have to indicate the card, not doing so would suck.
Making unique challenges for 3 stars on each level, okay I am not putting in the hours making that content, I have done my time here, and each level already is a unique challenge, a set of 10 kingdom cards and a set of starting hands and sometimes some other tweak, once that stuff is programmed. And at the same time, there are achievements, they are not in the game but they will show up someday, there are a ton of them, one for every kingdom card in fact, giving you some special challenge, and those challenges don't need to be specifically accomplished in campaign games, even better.
Possibly stars could be something that necessarily meant beating the level twice? Beat it again without buying any of the cards you bought last time. That seems too hard to track - obv. you don't want to be locked in by your first victory, so it would be like, track every set of cards I used to beat this level, and well not happening. I wouldn't want to just have a flag per card and make it, 3 stars if you've bought every card on this level over any number of victories, because then I just try to build a fairgrounds deck every time, I am winning okay quick buy up stuff.
It's harder to win if there are more players. It could be, if you want to go for 2 stars / 3 stars, we add a player / 2 players. Then either I go for that game one again every time and am now just waiting extra for bots to play (honestly not so much on very fast, but if they are all playing attacks or Bishops or whatever it adds up, and not everyone wants to play on very fast), or the game makes me beat it multiple times - win once for one star, now you can choose add a player and go for two, and well, I am not feeling it.
Anyway maybe there is some good idea out there; as it stands I don't see something better than score. At first I thought, mix score and turns, but really that is just saying "okay if you 3-pile it with one extra Estate you deserve the stars," and man, what was that about replayability and more challenges? If you beat it that way and want the stars, beat it the lots of Provinces way. The low score victory is still there for you if you just want to get to the next level. It's simpler just having the one system. I could see saying "some levels it's score, some it's turns" in order to deal with stuff like Goons, but mixing them is more complex, and people need to know what the threshold is.
Which is another thing, the screen should say what the threshold is.