A friend of mine works multiple part time jobs and has a pre-existing medical condition. Not surprisingly, he supports government subsidized (though not free) health care for people like him: working full time but without benefits. Companies don't want to provide benefits and often fire a full time person to hire a pair of part time people: same work and pay, but no costs of benefits. But this firing/hiring trend is common in every recession: it seems strange to create a permanent agency to solve a temporary problem. Still, the multiple part-time jobs is a permanent fixture in the economy and is worth thinking about.
My knowledge of tax law is somewhat lacking, but my understanding is that firms give benefits instead of an equivalent amount in cash because (a) tax laws make benefits cheaper and (b) people prefer that warm feeling of someone watching out for them over cold hard cash. Besides, matters of mortality is not something people like to think about so there's benefit in having someone else handle it. But part time workers aren't paid enough for the worker to be willing to take such an income hit in exchange for benefits. Moreover, tax laws wouldn't motivate the employer and minimum wage laws would prevent people from working for just benefits.
It seems we're stuck. Either provide subsidized health care and suffer all the inefficiencies that come with moral hazard or let the working poor suffer and with it the costs of delayed care.
But suppose we re-wrote the tax laws so firms would get tax breaks for benefits of part time workers and created an exception for the minimum wage laws (or just got rid of them) allowing people to be paid an equivalent amount in benefits. That way someone working multiple jobs would have one job where they're paid exclusively (or partly) in benefits and other jobs they get cash normally.
There's surely additional complexities because I'm not familiar with all the details of the tax code. But it has the advantage of giving people access to greater health care without running into the strong case of moral hazard that spawns comes from universal health care. The only hitch is that a lot of politicians hung their hat on universal health as the only reasonable solution so a compromise in the tax code probably won't be enough satisfy their constituents.