Yes, very young children don't come up with the idea of being trans by themselves. That idea is put there by adults.
I'd like to see people who talk about 'trans' children use language which reflects what they are suggesting about these children. They are saying that these children are transsexual. 'Trans' is a sanitised term which hides what this is really about.
The word 'transsexual' is a better descriptor than 'trans' because we are talking about children being put on a path to having puberty blockers to arrest their sexual development, followed by opposite sex hormones and surgery to alter their sex organs and secondary sexual characteristics.
Why don't parents who put their children on this path and schools which are socially transitioning pupils talk about 'transsexual' children rather than 'trans' children? Do they think there's a difference between 'trans', 'transgender' and 'transsexual'? What is the difference?