I have found that even when a villager refuses to heal someone at first (or can't do it), if you are persistent, they'll do it on the 3rd or 4th try. Each villager is different, but they will all heal if you give them a chance.

I agree with Rock that having the kids heal gives them a jump start on their skills. However, I do my village a bit differently by having every villager trained in healing to at least a Doctor Trainee. I don't have any actual doctors. But as each villager turns Elderly, I switch them to Doctor which means there are many, many doctors all over the place. Some heal a whole bunch while others don't do it as much. Some heal so much that they become Master Doctors (remember, they didn't get switched to that skill until they were 65 years old!). Anytime a villager gets sick, I have a whole bunch of doctors who run to heal the villager. Only the villager closest gets to do the healing.

So, yes, every villager can heal another villager regardless of their skill level or even age.