ETA
We just looked away for a moment, I was watching what I was doing while driving, it wasn't my fault. I didn't see anyone around that didn't belong...
All excuses when something happens to kids.
The mom is sitting by a friend and someone kidnaps their kid. They just "looked away for a moment". A babysitter should be in the sand and on the equipment playing with the kiddo. Not sitting on their hiney hanging with their friends.
They get in a car accident and it was the other guys fault. Baby still dead perhaps. Could be because girls were chatting and listening to music loudly and not paying as much attention to other drivers as when they're alone in the vehicle.
Things happen when we're with out friends and we are talking and looking at pics on our phones and not watching the kids all the time. I sit at the park facing the play equipment. I sit with my back facing away. If I'm with a friend they sit with their back to the playground so I can see the kids 100% of the time and I notice if even a new car pulls up. I watch for strange actions on adults parts. I expect something to happen so I am hyper aware in public places. We can't be too careful. At home the kids get a lot more freedom but in public they are quite restricted. The kids are in the basket until they are much older. I don't allow them out of my arms reach when they do get older. I don't give anyone any opportunity to grab one of my kids. They have not idea why they stay so close but I don't take chances.
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If you are both adults then the mom has no business saying who can come to this adults home when her child is present.
If she is babysitting this child in that child's home then you are not an invited guest of the homeowner and in fact trespassing. So stay away.
If she is babysitting the child in her own home she needs to check with the states laws about running an unlicensed child care business because she could be breaking the laws.
When I was a nanny (That means I went to their home all day to watch the kids) I took the kids anywhere and everywhere I wanted. They trusted me to be their kids supervision. If we planned a trip out of town the parents of course funded it and provided the transportation. We took all 8 kids I watched to the OKC zoo on one trip. I took along another adult and paid their way/food so I'd have another adult helping.
I was an adult with children when I was a nanny. I had many years in child care by that time also. I was someone they could trust and they knew it.
If you are not an adult who is a responsible person that is living on their own and the babysitter is not the same then this mom is absolutely right. Her babysitter/nanny should not be doing stuff without her full consent.
As for her pay. She'd be paying a child care facility about the same amount as she's paying your friend. I think the money is good.