Ultimately a classics degree only limits her job options in the short term. An engineer can go be an engineer, or transition to anything a classics grad can be, assuming both can sell themselves equally well. (if we are assuming high quality white collar jobs that pay well and have good career prospects)
Of course, college brand name greatly contributes to "selling yourself", so an expensive classics degree from Harvard is at a significant advantage over an engineering degree from some no-name university for non-engineering roles.
This 100% - you said it better than I did.
It's the underlying concepts and meta-skills that stay with you, and most "knowledge worker" jobs - not to mention real life - depend heavily on communication and learning to master complexity.
Even in CS I'd say that while the technology quickly becomes "obsolete" the general concepts and ways of thinking (not to mention underlying math and theory) last a lot longer.
If writing well is to do it simply, concisely, and interestingly, then I don't think anyone tried to teach me any useful skills related to writing after eighth grade. I learned more reading "On Writing Well" in a bookstore waiting for my dad to buy a book in my 30s than I learned between 9th grade and the end of my B.S.
I don't remember anyone teaching any classes on how to learn. Would be incredibly surprised if the classics had changed that much since I went to school. There are skills that are useful to learning how to learn like how to use Anki, interweaving, visualization, chunking, etc.. but all of that information comes out of psychology research so I'd be surprised if it was taught in any of the classics.
Maybe I am deluding myself because my daughter has decided to get an expensive classics degree, but the most valuable skill one can acquire in college is how to communicate, particularly how to write well. The second most valuable skill is learning how to master complicated material. Each major teaches these skills in a different way, but the value of learning these skills is basically the same however you accomplish it. It might be easier to get a high paying job right out of school with a CS degree, but the ability to communicate and the ability to learn will still pay dividends long after your technical knowledge is obsolete.