You may be right about that being what people think, but the reason would be right wing alarmism about “socialized medicine.” None of the single payer plans I have read about propose nationalizing either hospitals or doctors. This would seem both disruptive and generate more opponents in a battle that would be hard enough already. Do you have an example of a legitimate proposal for nationalization, one from a reasonable source? So I don’t think it implies it at all. When the AMA was fighting against Medicare, they called it socialized medicine also.