The idea that there is a male way of thinking and a female one is what feeds the idea of 'gender', and leads to people believing in 'the wrong body'.
I can understand little girls wanting to be boys - not because they have 'male brains', but because boys are (or used to be) given more freedom, and encouraged to do things that most children enjoy, and girls were not. How often do we hear women citing the fact that they enjoyed climbing trees as evidence that they were 'tomboys', for instance? Isn't it just that a lot of children want to play outside, but wearing dresses makes it more difficult, and many parents value(d) 'daintiness' in their daughters and 'toughness' in their sons? Similarly, not all boys enjoy, or are good at, football. This doesn't mean that they are 'in the wrong body', just that they don't enjoy football.
If we can get past the idea that certain likes and dislikes are particular to one sex or the other, children could just be who they are, and with luck, a lot of this nonsense would stop. It is regressive, and IMO the wrong way round. Society is constructed around the idea that men earn money and women look after children. Everything from the tax system to communication styles supports that construct, and children absorb it, too. Boys learn that being assertive and confident get them praise, whereas girls are often criticised for being pushy and 'bossy' when they do the same things. Therefore if a boy isn't confident and dislikes sport, or a girl is fed up with being called pushy and is good at science they can jump to the conclusion that they should have been born in the other sex.
We were moving away from that sort of thinking until the trans lobby reversed it. I did A level sociology in the 80s, and we spent ages looking at gender and how it was learned through the way people speak to boy and girl babies, and which toys toddlers were given. By the time I had my children in the 90s things were much more equal, but now it seems worse than ever.