Most children have shared experiences of watching certain shows, movies, etc.
They probably have other experiences in common too - jumping in puddles, playing in parks, learning to swim, participation in scouts or guides, maybe reading certain series of books, etc.
But by about age 10/11, their horizons tend to expand to the wider world, and not having exposure to media will result in a feeling of exclusion from a club. I'm not saying other children will necessarily exclude them; I'm saying they will feel they're missing something.
When a child's home home life and the life of her peers are so far apart, there is a risk of damage to the relationship with the child's parents. The parents run the risk of coming across to their children as people from a different culture, even people brought up in another country. There are limits to the extent that parents should be counter cultural, imo.
I brought up my children in the US, where I moved in my 20s. If I had kept them from watching TV, the only cultural exposure they would have had in their own native country would have been their experiences of school and participation in sports. They would have enjoyed books, certainly, but they would still have missed a significant amount of American humour and wouldn't have the cultural capital that makes small talk possible - a soft skill you need in life, both in your career and among friends. It's how you connect with your peers and colleagues.
Fwiw, they learned the song 'Alice the Camel' from watching Barney. I hadn't heard it before, so they would never have heard it from me. One of my DDs told me her high school swim team used to sing songs from Barney and Sesame Street on the bus to and from meets. In ironic teen fashion of course...
If I had never watched any American TV as a child in Ireland, I would have been quite adrift in many social situations. I wouldn't have understood the many references to old TV shows that pepper speech, the same way that people who don't read much don't get quotes from the KJV or Shakespeare or P.G. Wodehouse, or people who haven't watched Fawlty Towers or Monty Python would miss many a famous line in conversation.
I would not want a child to feel they were a stranger in their own land.