I have always told my daughter that if I look after the children, I need to be able to tell them off if necessary, I don’t make a habit of it but I will do it.
My older daughter is not local, so regular childcare wasn’t an option, but I think we would probably have clashed pretty quickly, when I used to have them in the holidays, there was always some drama, like you, one involved inappropriate television, it was on CBeebies!
I usually stood my ground, because I felt she was petty and unreasonable at times, but still led to a bad atmosphere.
I also think that you need to be trusted to make your own rules regarding screen time, meals, and activities, obviously not doing anything you know won’t be approved of, but it has to work for you as well.