Well, for one, there's a greater chance that players get injured when more games are played. If the series were over now, there would be nearly a 0% chance that a player gets injured. It also gives players who are playing with nagging injuries less time to recover if it goes the distance.
You also have to look at it in terms of amount of games played in the last x number of days. As the 2nd round goes on, the team that defeated their previous opponents quicker will have played less games and had more rest in that time frame. For example, let's say that we beat the Jets in 7 and the Preds beat the Stars in 5. If the 2nd round series goes to 7 games, the Preds will have played 12 games in something like 30 days. The Blues on the other hand would have played 14 games in 30 days, nearly a game every other day for a month. As you get into those later games, especially with travel being back and forth after game 4, the team that has played more tiring and physical playoff-hockey is going to start to feel it.
I will say that the team coming off that huge game 7 win is going to have the momentum and "guns blazing" advantage in game 1, but I'd rather have the advantage in the later games as opposed to game 1. The first game of a series is always the least important. Look at the Jets last year after they beat the Preds in game 7, coming out and beating the Knights in game 1 and then losing the next 4. Or the Blues in '16 after the game 7 win against the Stars, we won game 1 against the Sharks and then got shutout the next two games.