This is a fascinating thread!
I remember as a teenager being highly offended when others often referred to my best friend (5' if she stretched out her neck) and me (a smidge under 5'7") as "Mutt and Jeff". I envied my friend because she looked so feminine and delicate while I usually felt huge and ungainly next to her. Looking back, that comparison, along with my IT shape, is what probably formed my perceptions that traditionally feminine dress, pursuits, and behaviors just weren't for the likes of me.
My height, though, meant I could look most men in the eye if I wore heels, and, I suspect, led me to spend more time in male environments than hanging out with girl friends. My height and body build was a privilege which I could exploit in sports and the workplace, although at the expense of downplaying anything that veered towards the feminine. So, yes, height for a female can be a privilege, but it can come with some hidden costs.
So I can kind of relate as to why Kate wanted to be taller--if you renounce ever being cute and dainty, might as well grab your heels and leverage your skyscraper tendancies.