I don't really agree with the biological sex thing, as the rules of how reproduction works are generally pretty straightforward and pretty much the same across most of the mammal species. But it's beside the point.

My question is specifically how do we diagnose gender dysphoria. Is it just based on the fact that a person might express himself in a way that society associated with one gender or another?

That would make sense to me. For what it's worth, from my observations of this community it appears to me that at the very least a significant part of self-{identified,diagnosed} transgender population seems to have an unrealistic, hypersexualized portrayal of what being a woman means, most likely influenced by pornography.

But why obscure and attack the concept of biological sex then? Doesn't that only reaffirm that what we've been doing this entire time is a right thing to do? Imposing the correct gender role (or as you called it, assigning gender at birth), seems exactly like the way to go, as it lowers the chance of child developing gender dysphoria and having all the problems associated with it.