I have been to Canada many times for business at my last job. Many of these trips were to medical centers or research centers dealing with the healthcare industry. On every trip, the topic of universal healthcare came up. The consensus I got from the residents there was that the system is ok, as long as you don't get sick. If you need to be seen then plan on having to wait on a list for a long, long time sometimes months for a routine checkup. If you need a specialist or some special procedure, this could even take up to a year. Most of the people I talked to, who were professionals and could afford it, went to private providers for their healthcare. This means that they were paying for it twice. These are the reasons why I, and many, aren't fond of the idea of government run universal healthcare.
I have never had to wait more than a week to see my family doctor. Even if they are busy (like you can't get an appointment for a few days), you can always go down to the clinic and see someone within a few hours (and the wait is only that long if the line-up is extremely long). My family doctor works in the clinic so I get to see him most times.
Now to speak specifically to 'routine check-ups', that has to be the weirdest criticism I've ever heard. My wife/I's 'routine' (annual) check-ups are booked when we leave the prior one. That's what routine means, that you know of it well in advance. Essentially, your criticism makes no sense. My daughter's routine check-ups are booked in similar fashion, and have been within a day or two of the recommended date each time (one month, two month, four month, six month is next). If my routine check-ups are booked a year in advance, then what difference do wait times make?
Waiting a year to see a specialist for something is a myth bordering on a lie. You would think over 25 years of living here I would know of at least ONE person that this has happened to, but I don't. Now, I do know of a person who went in for a routine colonoscopy, and within a few weeks had his prostate removed. Doesn't seem to fit your narrative though.
As to the private insurance, I'm a professional, and work with professionals who make far more money than I do, and don't know a single person who uses private health care for covered services. As an extension of that - I don't even know of a place where you're allowed to pay for covered services in Ontario, outside of CT scans / MRIs (and even then it's only after hours). What provinces were you in when people said this, and were they travelling to the US for their private health care needs?
And even if this stuff WAS true, private health care doesn't solve those problems for anyone without insurance. What's a worse scenario - waiting a month to see an oncologist, or never getting to see one?
Here's a great final test - let's see if a single Canadian comes in and corroborates what you're saying. Is there anyone on this board who pays to see a family doctor? Who's waited a year to see a specialist? Who's waited months for a doctor's appointment? Who wants a full private system over what we have today?