The decision of what is a sensible course of treatment is between a patient and a doctor. Since you need a doctor's prescription to get birth control pills, it does fall under health insurance coverage. Sorry, just does.
The person paying for it is rightly part of any discussion. But I like your logic:
1) Government prohibits birth control without a prescription
2) Therefore, it's OK for government to force insurance companies to provide free birth control.
That would be a circular argument. Or for liberals, a circle jerk...
Government regulates it because it is a drug.
Perhaps you need to educate yourself on to WHY we have an FDA and why the government regulates drugs. This information has been deviously hidden from you in things called "books".
Short version, anyone could mix snake oil and cocaine and sell it as "Doctor Quacks Miracle Medicine", which usually got people high and provided no medical benefit.
There's a good reason why you need a prescription. Some women have bad reactions to the pill, so a doctor is needed to find what the best solution is.