It's definitely not appropriate for a kid to be doing drag on TV, in clubs, etc. Kids playing dress-up in their own homes is fine. Given that the mother's first thought was that the kid wanted to "become" a girl I'm not surprised things are going this way, and I think the main focus here should be how she came to that conclusion rather than just thinking that she had a kid who liked playing dress-up, which is what would have happened 10 years ago.
The whole industry of children on reality TV etc performing in sexualized costumers for the entertainment of adults is grotesque (and I mean the beauty contest kids too, and the ones who do dance).