Bollocks. I use my health insurance (publicly funded - I'm Canadian) for oil changes and checkups, and Canada still manages to spend about half, per capita, on health coverage as the US.
Not only that, but most employers offer supplementary insurance of one form or another to provide for the things not covered by public insurance. Because they don't have to worry about most of the Big Things (except dental and usually optometric), it's far easier and cheaper for them to offer some of the smaller things (massage, orthotics, etc.).