It would depend on the rules. If the government system says you can't work outside the system, then no they couldn't do it.
What I am saying is that there is no reason to believe that health care delivered by a for profit system will be anything but more expensive than a non-profit system, a single payer system for example. This video gives us no reason to doubt this.
The video really doesn't address that question and I don't want to go down that rabbit hole.
Unfortunately that is the only reasonable thing to discuss, costs. That is the failure of the US's health care system, it costs too much.
It destroys our competitiveness in foreign trade. We spend twice as much as the average developed nation on health care.
It wasn't always that way. Someone posted graphs of the health care costs of the developed countries from 1980 on. You can look at the graphs and see that if you project the curves back to 1970 that they converge. The US spent about the same as the other developed countries on health care.
In 1970 we had a largely non-profit health care sector. Hospitals were owned largely by charities and local government. Insurance was provided by the large, non-profit, community rate based companies, primarily Blue Cross-Blue Shield or by the government, indigent care, Medicare, Medicaid, the VA and the military health care system.
You have to look at what changed. It was the ever increasing conversion of health care into a for profit business. Hospitals were the first, for profit businesses bought the hospitals from the charities and from the local governments.
Then they broke up Blue Cross-Blue Shield and for the most part turned them into profit making enterprises. Before Congress broke up Blue Cross-Blue Shield Only about 5% of the health care insurance was issued by for profit insurance companies where they were at risk. Their main business was administrating the health care for large corporations, where the corporation was at risk, not the insurance company.
And no the increase in cost wasn't because of the huge increase in government regulations. The huge increase in government regulations was because of the increase in the penetration of the profit model into the health care industry. Who needs more regulation, the Sisters of Mercy running a hospital or a for profit business running one?