The problem is that if you use Shamir's Secret Sharing and there is a majority of dishonest players, they can collude to reveal your seed before picking theirs (giving them the ability to effectively pick the game outcome). There are indeed solutions using a trusted third party.
Isn't it that the seed or the part of that secret is only going to be given to the players and they could just verify it and not change the outcome itself?
From what I understood with Shamir's secret is that you have a secret and split it into other secrets with having the same size as the original. You cannot have the whole secret to be reconstructed without all the parts right? Or having the minimum threshold which you will define.
Let's say every game, you have an arbiter and 3 players. Making which threshold be 3. I think
n - 1 with regards to the number or Total Players would be sufficient. n is the total number of players.
Even if you have 2 dishonest players, the total secret cannot be reconstructed unless you have that piece of the arbiter.
Giving power to that arbiter could be the crucial part and just like you included, trusted third party would be allowed.