1) Health. Agree that talking to a doctor -- or a nurse, or a midwife, or a well-trained community health advocate -- is quite useful when understanding the practicalities of taking hormonal birth control. Talking to someone helps people understand this; I think you may overestimate the efficacy of an MD per se here.
2) Insurance as gatekeeper. The narrow windows to get a limited supply, whether you can get it locally or must mail order it or can't mail order it or can or can't get a refill -- none of this is medical. It's all insurance bureaucracy. I find no benefit to this bureaucracy.
Good education about the full range of options would be great (from pill to ring to IUD to condom). Universal health care availability would be a good mechanism by which to deliver that, it's hard to accomplish when it's all gatekept by insurance.
I got laid off recently & had a routine physical that took place 2 days after my health insurance thru Company ended, went anyway, got a bill for $552. This is for conversation only, haha, but the type of conversation one would need to get an annual renewal for birth control controlled by prescription.