[0] There's a distinction between "public" (gesetzlich) and "private" (privat) health insurance, but neither is run by the government. Public insurers are private organizations. The difference is in what is covered (all public insurers provide essentially the same cover), and how they are funded (the cost of public insurance depends on income, while private insurers can set their own prices). There is an income threshold below which one is required to take public insurance; private insurance is seen by many as kind of a scam. Private insurance is generally cheaper for young people, but one can't easily switch from private to public, so you'd better be sure your income will continue to be high as you get old and premiums go up.
And they do have single payer in a sense. The consumer pays the doctor and is reimbursed by a health insurer. Something that pretty much seems like it might not work well in the US, without very tight reins on the insurer reimbursement timeline and approval.
Well as someone from Canada, living in a provincial capital, who can't get a doctor (for years), I'd say I agree with the OP.
I guess the family doctor waiting list I've been on for two years and counting is simply a "bogeyman".
No, but it's a problem coast to coast, apparently.
>every time I actually try to access healthcare it seems to work much better than my friends led me to believe it would.
You sound healthy. Not everyone is so lucky.
Also, it seems that your condition isn't life-threatening (considering you're posting here and all). While the average, generally healthy, person will likely need to wait longer, they at least have access to care in developed countries outside the US.
The final point I would add is that private healthcare is still available, even if only available in a different country. I think that's a reasonable trade off- most people can get care readily, while those who need more can pay more to get more timely care.