I think for the math to work you have to consider the bigger bulk of work that could be eliminated by universal healthcare: the payer industry. Insurers, PBMs, and all the smaller sectors that support them. The utility provided to society is fairly low — basically, just selective claims denial.
Any model that gets rid of these frees up a huge swath of capital and work from society, and can use that work elsewhere. Of course, that is easier said than done.
Truly. Aetna, United Healthcare, Anthem, etc are examples of industry that does not need to exist, and exist only to serve the status quo.
Each of those are companies worth billions of dollars that could be instead used to lower the individual cost of providing health insurance.
As you say, this would of course be unsavory in some respects as those companies employ a lot of people. It's not a very economically productive industry though, the main output seems to be consuming patients' and doctors' time, causing financial anguish, and causing stress among people as to whether or not if their condition will be covered.