There aren’t many colleges presenting a classic education. There are a few though. They teach the “classics”, true, but that isn’t the point. They provide a classical education in the sense you mean, teaching people how to think, and leading them to think about the big issues of life.
There are others. For some reason they tend to be catholic colleges, though not all.
Often, such colleges follow the "great books" model of education, wherein one reads the actual historical cultural sources, rather than textbooks that summarize such material.
Thomas Aquinas College is a splendid example of a great books institution.
Another splendid example is Hillsdale College. They do not take a dime of taxpayer funding. Happily, this means that federal and state educrats have little leverage to use against them if they do not comply with the educrats' prescribed orthodoxies. Which Hillsdale — a liberty-loving institution — actively opposes and resists. They actively reach out to the national community with their free on-line courses — check out the latest, "The Presidency and the Constitution." Then there is their free newsletter, Imprimis, which features wonderful writers from inside and outside academe engaging critically important topics. The November issue, now out and available on-line, is "Effects of the Affordable Care Act on Economic Productivity," by Casey Mulligan of the University of Chicago.
Then there's Hillsdale's Washington, D.C.-based Kirby Center, a sort of lobbying organization that vigorously promotes the Constitution and conservatism on Capitol Hill and beyond.
To return to our main issue: Under the great books model, one must actively do one's own thinking and analyzing, rather than just passively receive pre-fab material composed by "experts" and "pre-digested" for us. This is the progressive approach to pedagogy. E.g., you don't actually have to read Herodotus; instead you get a description of who he was and what he said.
Some catholic colleges and universities have become quite progressive in recent times. You can pretty much ascertain which are the progressive ones by finding out whether or not they use the great books model in their humanities departments....
Anyhoot, just some thoughts, FWTW. Thanks so much for your keen observations, dear brother in Christ!