I think it's a great idea. But, as always, determining what is on that bookshelf is a tough one.
The Harvard Classics was a great attempt about a century ago [0]. 'Dr. Elliot's 5 foot shelf of books' was what an undergraduate degree at Harvard in 1909 included. Harvard University President Charles W. Eliot put together about all of the great books that those Gilded Age scions were supposed to digest.
Personally speaking, the greatest thing about the Harvard Classics was the 15-minute-a-day Reading Guide to the corpus. I went through it all a few years ago, and man, that was a really good idea. Like, I like Shakespeare now; he's really good! And I read through that philosophy finally, and that was really good too! And, lord, the poetry; I never really liked poems, but now I really love them. Most days were like that for me. I'll fully admit though, there were some real stinkers in there. But only once in a while.
If you'd like to read along too, you can download scanned versions of the books here: https://www.myharvardclassics.com/categories/20120212 . For example, today's reading is:
Vol. 2, pp. 31-43 : Plato's Crito -- Socrates unceasingly strove for beauty, truth, and perfection. Sentenced to death on a false charge, he refused to escape from the death cell, even when opportunity was offered.
As an aside, I'd love for an updated version of the Harvard Classics complete with 15-minute-a-day type readings or other media. I really did look forward to that dose of mind-training every day when I went through it. Does anyone know of a good place for daily mind training that is like this?