Relly cool idea, but i feel like the random aspect is a bit too important. Maybe try to skew the random part a bit to make it less likely to have only one and two dices, or all dices of the same color (on six turns it's very likely that it won't balance out). I also would have liked a bigger board to have a bit more agency, i felt a bit constrained at times (but that's also part of the fun ;) )
Really think there's something here, would like to see this polished a bit further!