Only if the insurance companies form a cartel (in the economic sense). People will switch carriers to ones with lower premiums so the market forces direct costs down to parity.
Most costs are outside of insurer's control anyway, regulations prevent insurance companies from telling providers how to offer care as long as the care is medically necessary and the standard of care.
While it's true that a cartel would be the fastest way to raise prices, I don't think that not having one removes all incentive to try.
I think you're also ignoring healthcare networks. This is important for two reasons.
1. The kind of supply and demand works very well for modeling commodities, but the difference in networks means it's very hard to have two completely equivalent insurance products.
2. Insurance companies can incentivize hospitals to behave in certain ways by regularly pruning those who do not behave that way.
Also, most people get their healthcare from their employer. There's not as much ability to actually switch, unless you're so fed up that you're willing to switch jobs.
If both the insurance and the hospital earns money by raising the price we will get what we have today where insurance covered procedures are more expensive than none covered procedures.
Not if the supply of healthcare was increased (more doctors). Then they would be willing to sell their services for a lower price, and the insurance companies would be able to offer lower premiums, winning business from other insurance companies offering higher premiums.
Most costs are outside of insurer's control anyway, regulations prevent insurance companies from telling providers how to offer care as long as the care is medically necessary and the standard of care.