Gender dysphoria isn’t about believing “boys should like football” or “girls should wear pink.”
It's a deep-seated, embodied sense that one’s body or sex category is wrong. The distress isn’t produced by stereotypes; it’s about the body/ sex itself.
A child may hate the colour pink, prefer football to netball, and have no interest whatsoever in conventionally feminine things. Yet every day they may still beg to wear a pink dress and play netball—not because they enjoy those things, but because doing so allows them to be recognised by others as the girl they feel themselves to be. The pink dress isn’t the cause of the dysphoria; it’s an attempt to alleviate it.
So what we're talking about here is a little boy who hates his penis (I assume this is what you mean when you say it's about the body/sex). And because he hates his penis, this little boy thinks he's a girl? Or wants to be a girl?
Don't you think the first thing we should be looking at here is why this little boy hates this part of his body? Has he been sexually abused, for example? What is causing this little boy to (a) hate his body, specifically his penis and (b) think that this body dysmorphia means that he's a girl?
There is, however, a more philosophically interesting argument. If sex categories themselves had absolutely no social meaning or consequence, perhaps gender dysphoria would disappear because there would be no meaningful category in which to feel misplaced. I can see the logic in that.
If gender dysphoria is only about the body, and wanting to belong to the other sex category and the stereotypes of the opposite sex are just an attempt to alleviate the dysphoria (as you've stated in this post), then doing away with the social meaning or consequence of sex differences (which is, anyway, impossible) would have no effect.
Even then, though, people would still have different bodies, and many people with gender dysphoria experience profound distress about those physical characteristics themselves.
According to you (quoted above) it's not "many people with gender dysphoria", it's all of them.