I've sometimes had to put a villager 5-8 times at the food bin before I've seen the "your villager feels better after eating" line. But even when I do see that, I might not notice a change in the healing status. But after several iterations of the line, I may see a change, but that could be a coincidance with the healing that comes over time.

I have noticed that nursing mothers and kids won't eat at the food bin even when dragged there, so they are totally dependant on the "natural healing" if they are already in the weakened stage.