Is unbalanced "win" possibility in a co-op game bad?

I am working on a cooperative tile placement game where players are exploring a system of caverns trying to collect lost treasures while escaping from golems that chase and kill you.

I wanted to add a side "win" condition though so that players have reason to not always play in line with the best interests of the group (i.e. you get points for each treasure collected and golem killed and whoever gets the most points is the expedition leader, but still only if the team wins the game overall).

The only problem with this is that the chances of getting the most points are lower for the first player than the last. Turn order has all players take a turn and then after the last player any golems on the board are moved. The most common way to kill a golem is with rocks you find around the caverns where one hit knocks the golem over and a second hit kills it, but if it is not hit a second time before the golems' next turn, it will stand back up and require 2 more hits. This means that one common way golems are killed is two players working together that have each found one rock, but then it is the second player to hit it that will get the kill/points.

Now, that said, the first player can really get just as many points from collecting treasure, and there are other ways golems can be killed that would not favor a later player, so it's not the kind of thing where even most games the first player couldn't win, just that they are slightly disadvantaged.

So my question is, does that matter? Since it's a cooperative game anyway and it's not an overwhelming percentage of games that would be affected by this, my thought is that it's okay. And I also figure you can say the Expedition Leader from the previous game would be the first player in the next game to give others a slightly better chance of winning next time.

Anybody have any thoughts on this?