Interesting that many of the early steam mills used a steam engine to pump water into a water tower, which then drove a conventional water wheel to power the millworks.

Yes, until they were able to get smooth rotation out of the engines, the water wheel was a much smoother source of power. Especially for looms and weaving machines, stable, smooth power was critical. "Power in the Industrial Revolution" by Richard L Hills is a deep exploration of the history there :).