I used to wonder about this. The thing is, though, first and foremost, unless your estrogen levels are low for a female, how could you even think you had a hormonal imbalance to begin with? I DO think it's wishful thinking on the part of parents, etc to think that being trans could be fixed with a pill or a shot, that you could just magically feel okay in your own skin. Hell, I'd love it if that were possible! As much as I love how T makes me feel, I'd love to not have to give myself a shot every week, and have to worry about a painful and expensive top surgery! But unfortunately, there's just no evidence to suggest that's anywhere near possible.
Besides the fact that other guys have already attested that even the small amount of estrogen/progesterone in birth control has made their moods, etc go haywire while doing nothing to curb their dysphoria, you have to also consider the fact that when a hormone is in superfluity, it converts to another hormone... i.e., when there's too much testosterone in the body, as in guys who shoot steroids, the excess converts to estrogen and manifests in external feminine characteristics such as gynomastia. In women, or female-bodied persons, an excess of estrogen converts to testosterone... so really, if you took too much of it, you could end up with a beard regardless, lol.
Unfortunately, there's no quick fix to the trans situation. And as I've told plenty of my trans friends, you don't choose to be trans... you only choose whether or not to transition.
SD