Not unreasonable at all, though I'm torn between feeling like parents are often not the best person to help explore complicated feelings (even well-intended parents) and concerns about the lack of resources outside of the home with how overstretched mental health services and the difficulty in finding the right other people to help. It's rough for too many children, girls particularly in this regard.
If someone tells you who they are, maybe you should believe them.
I started talking about sex changes when I was about 7. When I was 9, I would have said I am someone who shouldn't have been born (I mean, if we view that women who do not want children shouldn't have them, then it's still objectively true, but I had more emotions attached to it back then).
I was of course a child who didn't have the language or other skills to discuss trauma and emotions, I was just using what media representations and other rhetoric I had been exposed to - the joys of being raised on soap operas, it was mostly weird shite. I was very lucky to get schools access to mental health care - I saw the school psych in elementary, had group therapy in middle school, and a mix in the higher grades.
We never talked about 'who I was' - I was a child, my identity was continuously in flux and impacted by those around me (as is also true of adults to a lesser extent). We talked about coping mechanisms, making choices on our thoughts, how people aren't really all that great at knowing what causes or even identifying emotions all the time and how to practice that & understand that just like our other senses, our emotions send messages but aren't always reliable.
I find the individualistic identity centered way culture is going on how to view people doesn't do much good and just because someone says they feel a certain way doesn't mean it automatically leads to a certain identity as defined by the culture the person happens to have been born in. My feeling threatened by being perceived as female doesn't mean anything about who I innately am.
You cant just ban the internet with teenagers. They have to learn to navigate it. You can put parental controls to stop them having unfiltered access to porn and 18+ sites, but you cant stop them talking about things.
Banning would likely be difficult, but parents can 'intensively monitor' it within their homes, especially for younger teenagers.
After many long conversations about issues my then-13 year old DS1 was having online, he asked us to remove his phone's web browser he found even parentally blocked internet access too much and we listened. He spent a few years only working on computers at home on a computer attached to our TV where everyone could see it. He was much happier and was perfectly able to learn how to navigate it doing that way. It's not really that hard to do, especially these days, and a bit of extra maturity really helped him so now he manages fine.
I had pretty much free range internet usage from an early age, joys of being a late 80s kid with a dad working in tech where a home computer with internet was a perk of the job, and 'they have to learn to navigate it' was my parents attitude that fucked us all up and led to my brother ending up arrested and expelled and barred from every school in the county. Even on non-18+ sites, you can talk about illegal shite that has repercussions.
I can't stop them discussing anything, but my children - particularly my sons - are in high risk demographics for radicalisation in a city which has known problems with it so yeah I restrict and monitor internet usage. There are so many other identity-focused groups out there other than gender-based that just love teenagers, inexperienced by definition, looking for community, understanding, and an explanation for their feelings.