the US is in a position of needing to dismantle the existing system before it can errect a new one,

That just isn't true, and it isn't how things are done, unless you are Pol Pot or similar.

The system of health care that existed in Britain before 1948 wasn't "dismantled", it was either incorporated into the NHS, or it continued, and continues, alongside it.

A major reform to the medical insurance system in the USA, which would mean that everyone was covered, and that it would no longer be possible for private insurance companies to exclude people and to rip them off is perfectly achievable, and does not involve "dismantling the previous system".   

If it caused a shake out of the private insurance companies, all to the good, and could lead hopefully to a situation in which for-profit insurance companies were no longer a significant part of the scene. Private health insurance in the UK manages very healthily without such companies.

There would be no reason for any upheaval in the actual medical provision - family doctors would continue to be famiy doctors, hospitals would continue to be hospitals. Changes and improvements could come on stream as the need for them was recognised.

The system that emerged would be different from that in other countries, just as is true in these various countries. But there's no reason it shouldn't be a perfectly good system, providing quality affordable health care to all, just as all these other countries manage to do.