> Do we add the average cost of employer-provided health insurance to US wages to keep the number comparable? What about those who self-insure?

Obviously yes, and obviously no. Just as we don't add the price of groceries or rent to wages unless the employer subsidizes those in which case obviously yes we must.

> What does the raw tax/burden tell you here in an international comparison?

It tells you how expensive it is to live there, relatively speaking. Obviously (as you note) that's only half of the discussion just as an expensive appliance might or might not be worth the premium to any particular individual. But you're wrong to dismiss it out of hand as you do. Just as it's disingenuous to dismiss the benefits of (for example) universal healthcare it is also disingenuous to attempt to derail discussion that focuses directly on total cost or effective tax burden or what have you.

What you are dismissing as a "nonsense number" of an "interest group" I would consider one of the most accurate and easily understood quantifications of total cost.