If you had excess hair growth before starting birth control and being on birth control has stopped it, then you likely have some kind of hormone imbalance, such as PCOS. You should see an endocrinologist about it since hormone imbalances can affect your overall health and longevity, not just how often you need to tweeze.
And sorry, but birth control is just synthetic versions of hormones that your body produces *while* pregnant. That's how it works, but tricking your body into thinking it's already growing a baby and therefore doesn't need to ovulate. Taking BC while pregnant would do literally nothing. Also, you kinda need to stop taking it to get pregnant in the first place.