I let them watch one show a day (20-30min). My kids are 2.5-3yrs. I limit it to a handful of shows that I choose from (Mother Goose Club Songs, Raffi song videos and right now Christmas classics).
I put the show on during lunch. Which is rather backwards I know but I don't like it on during play time as they stop playing and sit around watching. My DCG is a really picky eater and I have the rule that we stay at the table until the meal is over (my daughter eats lots and isn't fast about it). Since starting to put the show on DCG has started eating not only more but more variety. She always tells me right away "me no like that" then she eats the part she likes...next thing you know she is watching the show eating all the foods she does not like.
Now don't get me wrong, tv during meals is generally not the best, they don't talk and socialize and they eat more than they need. But I have found it useful. We don't have it on during snacks (or other non daycare meals) and I don't have it on during any other part of the day.
By having it on only during lunch they don't ever ask for shows during the day because they know it is only during lunch.
I will note that this has been the routine since down grading to just one DCG...when I had 5 we had zero screen time and the parents did indeed inquire about this. My current DCG watches tv 24/7 at home and they constantly give me long lists of her current favorite shows. They could care less if I use tv but I do not use it as I do find it negatively impacts their free play.