A big effect it would have is making ending the game on piles not make sense, ever. Because if you're in a remotely close game, if you spend a buy to end the game on piles, the other person will have presumably an equal turn, and not spend any buys on anything but green.
Makes mega-turn games really weird - you can't pull the trigger until you can buy up more than half the points, not until you can end the game on piles as now.
Actually, would give p2 a BIG advantage in mega-turn games! Because p2 can end the game on piles while buying just the minimum amount of green, whereas P1 would never have that option.
Would be interesting, but I don't think it would necessarily make the game more fair. I suspect that p1 vs p2 advantage would then vary based on the cards in the set.
Dominion is a short enough game that an easy way to to balance out first-player advantage is to 'play to X games, loser of the previous game goes first, have to win by two.'