I was in Hyder, Alaska in 1992. A buddy and I spent the whole day, chatting with locals. A different culture indeed.
Citizens of Hyder sent their kids to the school in Stewart BC. For small medical issues, they went to the Stewart hospital. For bigger medical issues, they drove three hours to Terrace BC. The Alaska/BC governments came to some business deal for these Amercian citizens to have medical services.
I took a look at a map of Yukon, with a population of 40,000 people. If the territory cannot provide proper service, then next big Canadian "city" is Fort Nelson BC. That town probably doesn't have any better services than the Yukon. So the next destination is Prince George BC or Grande Prairie, AB. It just might be cheaper for the Yukon government to make a deal with the Alaska government than to build and staff those services in the Yukon. So Yukon plates in an Alaska clinic is not a sign of a poor health care system. Probably a sign of governments collaborating to use resources more efficiently.
I don't where Americans get their information about the Canadian health system falling apart. I have had two serious ailments. Both were addressed in a timely manner. Otherwise I would be dead. Recently a friend of mine came down with meningitis. He was airlifted to a better hospital in Calgary. He has since recovered. I see my doctor several times a year. No long waits. No big bills. My "coverage" stays with me regardless of my employer.
Since I moved back to my home town, I have heard only two stories of Canadians going to the USA for medical services. Both went to the Mayo Clinic. I've never heard anyone going to Montana. Maybe there are few. But for sure 99.9% of Canadian medical cases are handled in Canada.
But your story of dealing with health insurance companies is a sign of a broken system.