Babysitters are teens yes, but they are also homeless NPC Service Sims. They don't have a regular household in town. Unassigned or not currently busy children (newspaper deliverers) and teens (babysitters) will show up at school once in a while, but they aren't obligated to be there and nothing bad happens if they never do. They aren't really meant to progress as resident sims are. Which is helpful and kind of sad at the same time. Many of them in my game grow up to become YA Service and Role Sims or they leave my game entirely unless I do something else with them.
Sounds like your sitter might have the Kleptomaniac trait. You can check on and change traits like that with MC Cheats. Or she may have just picked up the portable radio "by accident." You could look for it or transfer it out of her personal inventory using MC commands, but I don't think Service Sims actually keep anything in their possession for very long.
The "has to leave" notification might mean that she was standing right on the edge of the lot at the time. Or that your older sim didn't leave the lot entirely quite soon enough. But if she stays and babysits anyway then it's nothing to worry about.