I get it, it's there, but it's a small thing and I think the rule of awarding ties with unequal turns to player two does enough to mitigate it.
And bad draws, well, YMYOSL.
As for tourneys, any tourney worth playing in will have opponents play equal numbers of games as player one anyway.
Maybe first player matters more in some games than others but, hey, it's a card game, there's some randomness involved.
If you really want to make a tourney as fair as possible, you can play each kingdom twice taking turns as player one.
I personally don't believe it's that big a deal.
Usually the better player wins.
There are so many contentious points in this post that I don't know where to start.
First player advantage is not mitigated by player 2 winning on ties with uneven turns. First player advantage has been shown to exist in simulation, and every top player will agree that first player advantage exists in almost every sort of kingdom with player 1 having a win rate of 55% or greater assuming skill is controlled.
Your use of the YMYOSL meme is inappropriate in this context. Assuming that every deck strikes the proper balance between speed and consistency, there is usually a point at which consistency has to be sacrificed for speed in order to maximize win rate. YMYOSL also doesn't invalidate the fact that player 1 has more leniency than player 2 when it comes to not losing a split. This has been observed whether the split in question is a victory card or an important supply pile. Player 2 only has to have bad luck once in order to lose a split; player 1 must have bad luck twice.
A tournament with elimination is not designed to have both players play equal numbers of games as first player. You can't guarantee a set winner with an even number of games if all of the games end in a win and a loss.
The 2012 f.DS championship rules were that the higher seed was player 1 in game 1 and the winner of a game would be player 2 in the subsequent game. Therefore, it was important to win game 1 to maintain first player advantage through the entire set, and in order to claim first player advantage from a disadvantageous position, you had to win as player 2.
Requiring that players play a kingdom twice is not fairer. The second play of a kingdom is more informed and more optimized than the first play of a kingdom. Suppose that Mic Q and I play a kingdom twice, the first time with him being player 1. Since I'm probably now a substantially worse player, suppose that I pick a bad strategy and Mic Q picks a near-optimal one. Now in game 2 where I'm player 1, I can copy Mic Q's game 1 strategy (with appropriate modifications) and get a win in a kingdom where I was completely outplayed game 1. In this case, since I am very likely to lose game 1, I could just use that as an opportunity to sandbag, experiment, and note what a better player does so that I can use that information in game 2 when I'm player 1.
Finally, all of this is a dumb digression. In the game that I posted, the fact remains that by buying Moat, I took a match that would otherwise have ended in a draw or a loss and turned it into a win. This demonstrates an instance in which Moat was a good card.