Two points. First, Christopeher Hitchens argues that Hillary's "reform" was, in fact, a move designed by big insurance firms and received a relatively mild oppsotion from smaller guys in the insurance biz. So it was hardly a propaganda blitz that "killed" that initiative. Au contraire, the whole "initiative" was a scham never intented to be implemented as advertised.
Second, health reform involves two conceptually different issues - the cost-effectiveness and the universal coverage. "Single payer" or more generally - public insurance schemes are designes primarily to address the cost-effectiveness issue by reducing transaction costs that are significant in this business. It does not automatically lead to universal coverage - in fact the acclaimed public health care systems under state socialism were NOT truly universal - for example, self-employed were not covered. Moreover, not every procedure was covered - only those available in public health care facilities.
Universal coverage does not require a single payer solution - it is possible to attain by means-tested public subsidies of insurance premiums. That is, you buy your insurance from a market vendor, and if you cannot afford one - government subsidies will make up the difference between what it costs and what you can afford.
So it makes a lot of sense, from the Left's point of view, to make that conceptual distinction clear. As katha p. & others pointed out, changing the status in the insurance biz will be extremely difficult politically, and the left should focus their energies on issues that really matter to its constituents, i.e. working class.
I do not think that cost-efficiency should be of primary concern to the Left for a number of good reasons, chief among them being that insurance companies can take of that. Moreover, "government health care" has become one of the buzz-words that provoke a knee jerk reaction on the right - so fighting for a single payer system is not the best strategy for the left, except perhaps for scoring symbolic points in a kulturkampf.
A much better strategy is to focus on universal coverage - which as I have argued - can be achieved by institutional arrangements that are not limited to a single payer public insurance scheme.