One of the problems with the cost of healthcare is that a third party is in a two party transaction. That third party is government. So the two main parties: patient and medical provider, don't have to really concern themselves with cost. You also see this with college tuition. Introducing a third party is always inflationary. I think something like half of healthcare spending comes from the government already with Medicare, Medicaid, SCHIP etc. There is your third party.
Another problem is going to be supply and demand. If healthcare is "free", you are going to skyrocket demand. This happens already with Medicare etc. Supply will not keep up with that demand. There is no way. What usually happens is that cost goes up when demand exceeds supply. However, if the government tries to control costs, you will likely have quality go down.
And again, the simple question, if giving away birth control would lower insurance company costs, why haven't they done it before? Are they just that dense? Most who want socialized medicine say that insurance companies are greedy. So if they are greedy, why wouldn't they do something to increase profit?
Insurance is something that covers disaster. You get car insurance to cover an accident. Same with homeowners and life. Health insurance is now becoming something to cover every health issue because it's subsidized. Imagine car insurance that covered oil changes? The cost just doesn't add up. Healthcare in America used to be like that. It covered disaster, and the patient paid for other procedures out of pocket. The government came in, third party, and distorted the market. To fix the distortion the government just doubles down.
*This is my opinion, your opinion may differ.