do they use the hospital

They don't exactly use it, but it is good to quickly locate people who ain't feeling well.

In the early part of the game sick villagers tend to hang about in the middle of the village, just south of the fire. Once the hospital has been built they will head for near there, instead. Since the village is usually getting quite large by the time the hospital is built it's useful to have them congregate there, as the area round the firepit has a lot of traffic, and it can get quite crowded.