Data point: I know several women who don't have issues with seeing a doctor in general, but do have strong issues with the particular type of doctor visit that's historically been required to get a prescription for birth control. Those visits can be unpleasantly invasive both physically and emotionally, when there's absolutely no medical reason for them to be that way. So being able to skip that uncomfortable visit is a big deal. The first time I heard a story of "so, yeah, I basically have to let this dude stick his hand up me once a year in order to keep my prescription", I was horrified.
Women should go to a doctor around once a year anyway just to check if everything is allright. Depending on a type of birth control those visits are even more important. There are reasons for them...
