The universe is a closed system, and entropy always increases, so the universe will eventually wind down and stop when it reaches maximum entropy.

It isn't quite true. The volume of the universe is constantly increasing, at an increasing rate even, and the maximum entropy of a system increases with the logarithm of the volume. Since maximum entropy keeps increasing, the universe can't reach maximum entropy and stay there. Yes the universe is moving towards the final goalpost, but at the same time the goalpost is running away.