Seems that is a feature of the plan. If you aren't seen on time you won't consume healthcare and the insurance gets to keep all the pretty money paid to them.
Similar things happen with the 4 hour target in A&E: if you're in hospital but conscious and not bleeding, expect to wait at least 3 hours. Having said that, I've twice been in for "this is either nothing or an early warning of something very deadly", been checked over, and of course not had to pay anything. That risk assessment looks very different if it might cost thousands of dollars.
The NHS is completely fantastic at bread-and-butter medical care: diabetes, asthma, minor injuries, obstetrics (it is insane that anyone ever has to pay for this), handing out antibiotics and contraceptives, and so on. The complaints people have tend to be about long waits for non-urgent surgery (anything up to 18 months), and the refusal to pay for marginally effective very expensive cancer drugs.
The only charge is $10 to get your prescription filled, and even that is waivable in poverty. On that subject I have noticed that US pharmacies stock a lot more OTC stuff, presumably due to the difficulty and expense of getting prescriptions. They'll even let you buy dangerous things like 500 paracetamol at a time.
If you mean that in Canada you're never ineligible for your physician because you changed insurance providers, that's basically right (except when you move between provinces, in which case you need to change physician anyway). So the whole problem cannot happen in Canada (nor in the U.K.).
I'm not saying everything's roses and perfection in commie-land... I'm saying that the U.S. system is just obviously horribly worse than all the socialist implementations.
(Well, okay, let's be fair... the U.S. system is probably better for the rich. When I was living in California, and had a medical incident, I got VERY good treatment... for which my employer's insurance company paid a HILARIOUS amount of money, and presumably charged my employer commensurate premiums. I mean, no-expenses-spared levels of treatment. I think Canadian doctors would have been slightly less cavalier about running up the bill.)
Meanwhile my friends in SV just whip out their credit cards, pay a small deductible, and get seen immediately.
Single payer isn't all sunshine and roses.
In the third world country, I had to have a very niche test that could only be done at the largest hospital in the city, a government owned hospital. It was a terrible experience. The corridors were packed with sick people sitting and waiting on the floor, it was hot and the ventilation was poor. I had to wait for hours to be seen while hoping I would not catch some disease while feeling incredibly sad for those around me that who must endure that hell regularly while for me it was just a once off.
In contrast, in the first world, the hospital care was just as good as the USA but without dealing with a billing department(can't remember why I was there, might have just been to see a physician for a cough). But from what I was told by a doctor and nurse I befriended while I was in the country, everything was done on a government budget. You want an MRI? If you don't have insurance, the expense comes out of the physicians allocated budget. What happens if the doc has expended the budget? You wait until the next budgetary cycle. Want a procedure done (that is deemed non-urgent)? Its free, but you need to wait months until there is space for you. Non-urgent includes cancer that won't kill you ... a friend's mother waited 6 months to get her stomach cancer removed. She was uninsured and it was free. Good deal (the surgery was something like 12 hours long with multiple surgeons working in shifts!) but they knew that if probabilities did not work in her favor, the cancer might have spread before they could remove it. They decided it was not worth selling the house and other assets to pay for private surgery. It was a good call in retrospect, but if it were my mother, I would have sold my house.
Another friend had a benign brain tumor. The doc was very reluctant to send him for an MRI. Three months later he got one. They instantly sent him for brain surgery. Removed it (for free!). But had it been malignant and spread, the time it took for the doctor to decide to use the MRI might have killed him.
The doctor friend I mentioned above, would tell me government bureaucrats would fight him on his spending, analyzing each of his patients and second-guessing his expenses. He hated them with a passion.
I remember only reading complaints about the UK system until ACA began to be debated, then suddenly it was the benchmark of great healthcare.
tldr; socialized medicine is great, and I would definitely support it, but don't fool yourself into thinking it's all that great. The upper-half of the middle class would still take private insurance and would still prefer being seen at private hospitals where they get the best care no expense (or bureaucrat) spared.
Things changed significantly for the better after Labour came to power in 1997 and began to significantly increase the budget for healthcare. I remember long waiting times for non-urgent care as a constant item in the news growing up in the 80's and 90's. With the increased funding it largely ceased to be a problem.
Currently the UK spends about the same as the US on public healthcare (a little under 8%) but gets full universal coverage free at the point of use. Selling your house to pay for healthcare in the UK isn't unheard of (desperate parents taking their kid abroad for experimental cancer treatments) but it's nothing like as common as it is here in the US where medical debts are the commonest cause of personal bankruptcy.