And that's one of the major arguments against public health care in the US: that the Canadian health care is so bad that Canadians have to come to the US to get MRIs.

I'd argue that it makes more sense to modify your system to make private heath care more available. I'm sure those people that travel to the US to pay for health care would much rather pay the extra expense in Canada than the US. After all, the problems you cite with the private insurance in the US drawing Canadians are the exact same problems that the "banning" of private insurance in Canada was trying to avoid.