***Sigh***. Here I am about to disagree and be unpopular again.
No, I don't think the problem is that gender is assigned at birth. For 98% or so of the population, their gender is simply not a problem.
The problem is that people don't understand gender, and aren't aware that there are people whose very brain wiring drives us to a gender that is different from our birth gender. IMO, the problem is lack of education about gender and lack of understanding that there are some people who just can't live as member of the gender matching their birth sex.