I don't think "anti-trans moral panic", and the analogy with gay, is a completely fair way of categorising concerns on this issue, for several reasons.
Yes, there will be some people who object on principle to the existence of gay people and the existence of trans people. In their case there is a parallel.
But the critical difference between gay and trans for a lot of parents is that gay is not irreversible and trans can be. These parents are not phobic. If their child is gay, that's fine. If they later realise they are not, that's just water under the bridge in the same way as having had straight relationships that didn't work out.
If they feel they are trans, and later realise they are not, it might just be water under the bridge but there is a strong possibility that they might clinch the matter with surgery before they realise it wasn't right for them. Parents worry about that, even if they are not "anti-trans". They also worry about the physical effects of hormones and surgery, even if their child's trans identity turns out to he stable.
And I think it's unwise to dismiss the idea of social contagion and assume that nobody will take the trans path unless it's innate to them. Social contagion isn't a myth. It's absolutely a thing. Human beings learn from each other, copy each other, mirror each other without even realising they're doing it. It's the driving force behind changes in language (the organic change that happens over time), behind fads and fashions (advertising can lead the horse to water but can't make it drink), and much else. It certainly operates within groups, and can include outbreaks of damaging behaviours, like cutting yourself, or hooliganism. Some of the people who do those things would have done them anyway. Some, left to themselves, would not.
To assume that gender identification would be exempt from this normal human behaviour trait is naive, especially if particular identities have social standing within the social groups that matter to you, and if your present identity carries stigma in your eyes or other people's.
A young woman who has internalised the idea that being identified as a girl degrades her, *and" has a peer group who admire and support people who come out as trans, and rank cis women as second grade oppressors, has a lot of reasons to conclude she is really a man.