IMHO, the current awkward government-private hybrid insurance system is very unstable.
There are many folks still calling for repeal of ObamaCare and implementing (at least partial) free-market reforms, such as the updated Docs4PatientCare proposal.
Others are sounding a drumbeat for a full government-run "single payer" system. (Here's a typical piece from NYT columnist Paul Krugman.)
The "single payer" advocates correctly recognized that this was politically unrealistic in the initial debates of 2009-2010, but hope that the political climate is more favorable to them now.
Likewise, free-market advocates correctly see immense dissatisfaction with ObamaCare and hope the time is right for their ideas to gain more traction.
I don't know which way the balance will tip, but I do think this will be the next big battleground of ideas.
In the meantime, ordinary Americans should ask themselves two important questions:
1) If the government's partial takeover of the health insurance industry has gone so badly, do we think a more complete takeover will do any better?
Also, Noam Scheiber makes the standard moral argument about private health insurance, essentially a variation of the "profits over people" line:
For-profit health insurance is on some level morally offensive—at least when it’s practiced the way we Americans practice capitalism. With a few tantalizing but mostly unrepresentative exceptions, the longstanding aim of health insurers has been to weed out sick people, and to weasel out of paying for treatment if they somehow get insurance, so that the companies could boost their share price [and] lavish income on their executives...Would a government-run health system really be any different? We've already seen that government-run insurance systems in Canada and the UK also aim to limit care (and costs) for budgetary reasons. The rationales used include "cost effectiveness", "quality", etc. But the end-point is the same -- they'll decide what medical care is considered "appropriate". And if you don't like it, too bad.
Americans should therefore ask:
2) Why is "for profit" considered inherently evil, whereas "government-run" considered angelic and good?
Whichever side answers these two questions best to the satisfaction of ordinary Americans will will the cultural (and eventually the political) debate.