You can get travel insurance for extended trips. I'm American and I've taken multiple 6-month trips abroad (usually after quitting a job). Backpacking just isn't part of the American culture.
That said, I've only purchased travel insurance maybe a half-dozen times out of probably hundreds of trips.
And trust me, you want a private hospital in some of those countries. Even the locals wouldn’t go to a public hospital if they had a choice.