Love the cat reporter, by the way.
When we lived in a different country Hubby would work shifts and be living in a camp in a remote area a couple of hundred miles away. The journey took him through quite desolate areas. I would track him (with his consent) as it was handy to know when he would arrive home. His shift would finish depending on the work and he would just get in the car and drive. It was also handy to know where each other was as it was an "interesting" county with quite a few challenges (I ended up in a taxi in the middle nowhere at 2 in the morning once.) Thankfully we don't need such back up now in the UK.
In the UK we installed a camera on the front and back of the house. We had to go to a funeral in Scotland and so hired a cat sitter to look after our brood for two nights with instructions that the cats should be in the house overnight. When we arrived in Scotland we gave the cat sitter a ring to see if everything was OK and she confirmed they were all safe, fed and in the house. Checking the camera we discovered that the cats were all having a party outside. We left after the funeral, a day early, and came straight home - only to find the cats hadn't been fed and the cat litter not changed. The cat sitter didn't expect us back until the next day. We called her to say we were home adding nothing else, and she got annoyed with us probably because she had been found out.
I agree totally that this babysitter is just the wrong fit for you. The babysitter is uncomfortable with cameras, which is her right, and you are uncomfortable leaving the children without, which is your right. Lots of babysitters out there.