You are, in a sense, getting it wrong. (by the way, I am not a Christian)
Single payer insurance is not the issue for most Christians. It is more about the inclusion of things in proposed single payer insurance that really have nothing to do with persistent health care as you put it.
Let's take abortion for instance. Abortion is not a health issue but a so-called woman's Rights issue. If a pregnancy is unwanted, then the ending of it should not be a Health Insurance issue. Abortion is an optional procedure that is embedded in social issues rather than health issues. Most devout Christians abhor abortion and do not want to be compelled to support it through a state sponsored single payer system. And that's just one argument associated with a Single Payer system, others include birth control related riders on a state sponsored single payer system. Birth control is another optional procedure that is mote rooted in social issues rather than health related issues.
Now add to this the "S" word (socialism) association with single payer insurance and the evil atheist based socialism becomes a religious battle cry.
What most Christians do not know is that the original socialist movements emanated from deeply Christian roots in the Northern European countries (Coops and Collective work Groups). The Atheistic association with socialism came from non-Christian influences like Marx, Hegel, Hershel etcetera.

ObamaCare in it's implementation strongly associated the things I mentioned above with government run Health Insurance. Thus the lawsuits involving the Little Sisters of the Poor (RC) and the Hobby Lobby corporation.

So, if a single payer insurance was offered that dealt only with actual health issues and not socio-political enhancements by certain agendas then I believe that the Christian resistance to single payer would crumble for the most part.