Not really, since people would switch to other health insurance providers who have lower premiums. The problem was not forcing everyone into the same insurance market.
Letting all the healthy white collar office people stick with their employer chosen health insurance means the leftovers are the poor and unhealthy which many insurance companies can’t handle the losses from since there is no health population to balance it out, so you have places that have no health insurance competition.
Even then, the people that make the most money in healthcare are the providers (doctors) and by a significant multiple, so increasing their supply would be necessary to bringing costs down.