So when they stay at grandma's on a school night because you have a work trip or something, it's fine for grandma to let them have ice cream for dinner, show them R-rated movies, and let them stay up until 4am on a school night because they want to? Of course your parenting decisions should be enforced even when you aren't there to enforce them, by the caregiver you've chosen. Obviously this kid's mom made a poor choice in thinking OP was a responsible caregiver, but yes the people you chose to watch your child should be supportive of your parenting methods.

Personally I think kids should have very limited screen time, because I think it's important to learn how to entertain themselves instead of constantly needing someone entertaining them. I think kids are a lot more independent and creative when they need to make a game out of some sticks in the backyard instead of doing whatever Dora the Explorer tells them. If I send my kids to a babysitter every day after school, is it totally fine for her to just plop them in front of the tv until I get off work, even though I'm against that?