A lot of kids want comfort and not cutesy stuff when they are sick even at age 3. Some parents also don't like someone doing the goo-goo stuff to their kids. Kids are now taught early to be on guard for strangers who use some of the words you suggest. If you acknowledge them by their name they may see you as less of a stranger.
Whatever works for you.
I'm glad to work in a small town. I know most of them anyway