I cannot comment on Canada, but will add a few cents worth on Australia - from my limited understanding the two health care systems are quite similar.
I can go to my doctor (GP) and the only time I will be charged anything is if I use their after hours service. With an appointment M-F, 8-5pm, I walk in see the doctor, get my diagnosis and walk out again and don't pay a cent. Same with emergency - 6 weeks or so ago my eldest (4yo) son cut his finger pretty badly and needed stitches. My wife took him to emergency and 3 or 4 hours later they came home, stitched up and didn't pay a thing.
I've been to the doctor for numerous things that needed more info (e.g. Xrays, blood tests etc.). In a lot of these cases, you get sent to a third party lab and you mostly pay for these. It is possible for these to get done for free in a public hospital, but you
may not get it same day as you would with the outside lab. In my local area, the General Practice Unit I visit is on the grounds of the local hospital and a great majority of the 'labs' you would need to visit are within a few minutes walk.
Surgery - lets say you damaged knee ligaments. This (again) *can* be operated on in a public hospital and as it is not life threatening is done on a first come first served basis and may require a significant wait to get in. The other option is via a private hospital and this is where insurance comes in (or your deep pockets) whichever you have. Currently, I'm not insured at all and in the (I suppose) low risk case I do have a knee reconstruction, I'll be waiting...
Other points like:
Maternity Leave - at this point there is no paid maternity leave funded by the government, though there is a push for company's to pay this out of their own pocket.
Childcare - when the carer returns to work, there are some rebates that are government funded, but they don't cover the full cost of child care (about 30% I think, and as I'm in a pub, I'm going to run with that figure
)
Family Allowance - we actually get paid to have kids - not sure how big the payments are but I think it continues until children are 16 (as long as they are still living at home). A portion of this is not means tested (meaning that it is not scaled based on income) and a portion is - no idea on the exact numbers though.
BTW, my income tax is about 1/4 of my income - I'm in the second top tax bracket (75,000-150,000) - you can see the rates here (though not from the pub):
Individual income tax rates