This is how the p2 gets an extra turn rule could change the game:
So you are building an engine as p1, and piles are running low. Oh no, you think; p2 might be able to end the game and buy an estate next turn. I had best buy a province and a market this turn instead of a market, a gold and a cellar (or whatever).
Then p2 continues to build their engine.
You still can't end the game despite your 1-province lead, because p2 can double province now, but you still can't.
So basically this rule would force player 1 to green early in a large number of engines, and I doubt there would be much difference in big money games.
This is no good at all; for one, we now just have a p2 advantage instead of a p1 advantage in a large number of games, that's useless. Additionally, engine games, where the p2 advantage becomes apparent, are the games where a second player's greater skill or play most easily allows them to overcome p1 advantage anyway, so it is not very helpful to give p2 an advantage in those games.