Any serious libertarian would say it (like most things) is absolutely none of the government's concern, at least with regards to consenting adults. There's probably a range of views when it comes to kids, though. Conversely, they wouldn't be interested in having the government policing the treatment of trans people between private parties, so they'd oppose things like legislation that protects them (or anyone) from discrimination in hiring, as an example.

FWIW, the author - Jesse Singal - is a writer I've followed for a while. I like him a lot - I find him level-headed and intellectually honest. I don't think he'd characterize himself as a libertarian rather than a liberal, despite being published by Reason here. He's just a science writer who ended up on the "trans kids healthcare" beat and has written about it extensively. I think he'd characterize his position as just "a lot of medical treatments for kids are being pushed on [in his opinion] flimsy science for [in his opinion] ideological reasons"; and he'd say that this is a scientific position rather than a political one. Of course he takes a lot of crap for this, and of course he's also attracted a fanbase of bozos for this. But his writing, generally, deserves better than either.

Libertarianism theoretically could go either way. Theoretically, do what you want with your bodies.

However, libertarianias as they exist tend to be socially conservative - somehow they end up agreeing with GOP position on social issues. In this case, convervatives hate trans people, so libertarians too.

I hope this view is "people are free to do whatever they want", because if libertarianism is only about ownership freedom, it would be the less consistent ideology ever.

Libertarians have a "my body my choice" position for things like raw milk and vaccines, and a "no you shouldn't be allowed that" position for abortion and hormones, because they've ended up on the rightwing side of the culture war.