The way I see it, someone who is playing slightly better might gain these small advantages more often than the other player, but not enough of them to convert the game.

That's because the other player is taking advantages back in his favor to even the board. You can convert a small advantage to a win unless your opponent doesn't claim take that advantage back from you in another way.

The logic is flawed in the sense that one player getting small wins on the board is plenty enough to convert a game. But in this case, those advantages were evened out by the other player.

I don't see how you can say one played better than the other in any way. That was the most even 12 games of chess i've ever watched.