No, sorry, no, it is not "broadly false." K-12 salaries enter at average 40k often with a requirement to enter a graduate program within five years. I don't see that teachers in most states have received substantial increases in salary over any considerable period. They are underpaid.
Compensation rates are not "surprisingly good" (surprisingly?). Both groups merit much higher compensation. Your subjective consideration of "well-compensated" may differ from mine and fair enough, but I find generally one's position is more an index of their political beliefs (or sentiments towards unions in general) than any objective standard of what is "surprising" ("a retirement plan? In this economy?).