I get the impression that the real problem with health care specifically is that we are keeping sicker people alive longer with more effective (and expensive) treatments, and this increased cost is not being reimbursed by valuable work done by those sick people

That's a problem. The international statistics suggest it's not the problem -- health care expenditures don't correlate particularly well with longevity at the high end.

Cultural tendencies towards proactive vs. reactive care might be responsible for part of this, but I'm unaware of any high-quality research on the issue. On the other hand, I haven't been following it closely.