True story...... when we lived in Iowa, one time, during the winter, my hubby was outside (it was dark, and there was probably a foot of snow on the ground). He saw a young child, maybe 4 years old wandering through the snow.
He got me, and we both went and talked to her, and got her inside. It was a small town, and most people knew everyone else, but we didn't recognize her, so we got our neighbor, hoping she would recognize her.
Well, the neighbor didn't recognize her, either...... we really didn't know what to do! (Today, we would probably call the police for help, but this was about 25 years ago, in a very small town.)
Hubby got the bright idea of back-tracking her from her footprints in the snow. They led a block or so to the back of a house, to the sliding door...... they looked inside, and there was a guy asleep on the couch!
Basically, there were 2 guys living in the house (none of us knew them..... they were renting)... and one guy's girlfriend came over, with her daughter. They wanted to go out, so they left the girl with the other guy, and he fell asleep!
The little girl opened the door and went out to "find grandma".
I shudder to think what might have happened if my husband hadn't been outside at just the right time!
You did not over-react in telling her that was inappropriate...... it was extremely inappropriate for her to take a nap when the other kids were playing.
I would really hesitate whether to keep her as a babysitter, if that is how she looks at her responsibility toward your children.