If that's the way it works out, sure, and who could blame them? If it doesn't work out that way, they'll vote for the other side in ever increasing numbers, for a great long time.
If it doesn't, anything that would increase costs to health insurance would then have to come from the already over strained general fund (buried and obscured), just to ensure the Democrats political survival. That'll require an equally unpopular action: raising taxes even higher.
Democrats now 'own' US healthcare. Any dissatisfaction with any part off the insurance side or any part of the medical treatment side is going land in the Democrat's lap, rightly or wrongly. Part of the function of managing healthcare is that you inevitably are going to have to deny coverage of somethings to some people, it's unreasonable to believe that everything can be covered fro everyone. Witness the UK NHS, short on money, they are now not covering some procedures that they used to, and the people don't like. Why should they? Government takes their money and delivers less of poorer quality. What's to like? Likely same will occur here after a number of years.
It's one hell of a bet to make with the future of your entire party, that government can actually deliver a satisfactory, one size fits all solution for everyone. Not a bet that I'd have made.