I concede the original point--it was an overeager, off-the-cuff generalization on my part, and it's not defensible. However, if it's possible to prove conclusively that healthcare in (as I understand it, quite literally) every other OECD country isn't substantially cheaper than that in the US, I'd expect the news to be fairly widespread as there's been no shortage of money behind fighting universal care in the US; I haven't seen it, and I've looked, but if there are data sets or analyses that indicate that's the case I'd be interested to see them.
To get back to the topic of the costs of education, everything a quick Google search turns up indicates that US education spending is on the order of double the OECD average, at every level; some higher-than-average spending may be justified but that seems extreme, and it does make it look like government involvement in other states is not driving their costs up the way that our system is, which looks to me to rule out the broad category of government spending/involvement as the culprit behind our out-of-control costs, though some subset of it (which might possibly not be a problem with some other combination of regulations or factors) may well be to blame. I simply don't see evidence to support the notion that government spending must drive up prices this way, nor even that it typically does.
Taxation aside, it's hard to believe that others spend more on education when they... don't. There are plenty of other differences to explain the tax disparity (and in the case of Denmark in particular, being at one extreme of the effective taxation spectrum, it appears you are correct that the disparity is substantial).