The sad lack in modern education is history. One reason our modern politics is so thoroughly screwed up is that a high quality understanding of history has been lost to the general population for a century.
I was looking hard to see if anyone had a glimpse of why greek and latin are important to education. You almost nailed it.
We've lost the art of teaching of how to think. The gentlemanly Greek and Latin were taught towards skills in reading texts, not in conversing to Joe Greek on the street about how he feels today; the pupil is then empowered to read many great and early works documenting the foundation and thought, and its progression, that form the fundamentals of our knowledge in philosophy, government, sciences and mathematics. Reading the literature of the time in the original source language conveys the subtext much more fluidly, thus enabling full comprehension. Individual languages are colored by the culture speaking it: Much is lost in translation. If you are to understand how to think, and achieve parity with where we have already tread in thought, then you need to understand first-hand how we arrived at the present knowledge, complete with the traps and tangents, not just the right answer. You learn how temporary some right answers are, giving you the humility and perspective to grow beyond the works of mankind thus far.