That would fix the final independence problem, but it would also require further changes to other stages of the proof. For example, this strategy would then me

That would fix the final independence problem, but it would also require further changes to other stages of the proof. For example, this strategy would then mean that when calculating the probability of a single bit being set, the hash outcomes are no longer independent, which means that a different expression would be needed. Additionally, from a practical sense, this variation might also be more costly to execute, as setting bits would go from a single memory access to a linear scan. I think it might be interesting to look into though.