The regions of the US with, as you allege, a “single payer system”, are not single payer. Single payer has to be from general taxation only.
Sure, single payer means something different in the US, but it will have the same funding issues, from which almost all of the problems within the NHS emanate, as the NHS. The important thing to understand about healthcare models is that what differentiates them is their funding — i.e. the fundamental difference between the Bismarck model, the Beveridge model, and the American model is how they are funded. Therefore, the fundamental different problems with each of the healthcare models are due to the mode of funding — hence why all single payer systems can be lumped together.