This is only an issue when both players are in extreme time pressure, and it could be a toss-up as to which player entered that extreme time pressure first and thus was longest on the hot seat.
This theoretical black edge is also marginally offset by white being able to make the first move with no concerns about what black has played yet and thus using, on average, less time for the first move than black would. Of course, it is a lot easier to lose a slight bit of time at the beginning of the game (when you have plenty of time to modify your time management) than at the end.