I don't which is why I support socialized single-payer medicine.
However, to address your question more directly, contractors are employees as well. It's not the job site's responsibility to provide health insurance, it's their employers. Contractors still have employers, you know.
Actually, your point reminded me that a great many employees don't receive health insurance through their employer. Those people are disproportionately low wage workers. Before Obamacare, they had no option but the very expensive individual health insurance market.
In your terms then, just let everybody be self-employed and contract them, rather than employ them. That's the same result I want, that people are responsible for themselves and making a contract with somebody to do some work for you doesn't come with any additional baggage. Just plain money for work.
That would be doable if the U.S. had a robust social safety net for all these contract workers, namely health insurance for those in between jobs. Right now relying on employers for insurance has been a terrible hassle.
However, to address your question more directly, contractors are employees as well. It's not the job site's responsibility to provide health insurance, it's their employers. Contractors still have employers, you know.