I don't specifically know the answer to that question.
However, the fact that you use the word assumed means that you think people are working off something other than knowledge of how the system works.
Can I ask if you are aware of place in Europe that don't have "single payer" ?
This, for example:
The Dutch really have it together on health care, they have a system that has
been proposed as a model for the US to emulate. In stark contrast to many other European systems, it’s actually based entirely on private insurers, rather than a single-payer or entirely national system. Yet the Dutch system is universal, has far superior rates of satisfaction with quality of care and access, and still costs a
fraction of what we pay for health care per capita in the US. How is this possible?
What is healthcare like in the Netherlands?