I don’t mean this in a condescending way, but that’s a great question. Lmk if my answer isn’t clear and I can elaborate more.
Single payer is not nationalized healthcare. Those are two different models with very different implications for funding.
Any business could buy up their downstream suppliers, just like the government could nationalize hospitals. But most governments (and most businesses) don’t do that.
In a single payer system, hospitals would get paid from one source, plus supplemental services that people buy out of pocket. What a hospital gets paid impacts what they can pay their employees. Ever heard someone say “why would they be an EMT for $15 an hour when they could make that flipping burgers?” It’s the same logic. Private companies lobby single payer govts so that they can attract employees, among other things they can do with money. More money makes running anything in a capitalist society easier.
If you’re nationalized, then the healthcare workers are government employees.