The difficulty you're having is that your villagers are weakened, not sick. Villagers can only heal another villager whose status shows "Sick" on the details screen. Since your villagers are not sick, the game treats what you're doing as a breeding attempt. Weakened villagers will recover by themselves over time. It helps to put them on a less dangerous job (like farming), so that they'll recover faster.
In addition to the on-the-job training that your villagers get when they're healing the sick, you can train villagers in healing by taking them to the cactus plants in the northern part of the village.
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Barbara
Unicorn
Last Day of Work