Why not read a bunch of books instead? If the point is to become a well rounded person, think widely and read widely, then getting a PhD isn't going to help you much. I think of PhD's as very driven, focused, able to take a lot of abuse, able to write and research for several years. I don't think of them as particularly well rounded. In particular, getting a PhD in Philosophy from Stanford after a BS in CS from Colum…
Now, the process of advancing the state-of-the-art is unusually hard and there exists a good deal of very good write-ups of the intricacies involved (such as "So long and thanks for the PhD") which requires people to be driven and focused. But, at the same time, before actually starting the research that gets them their PhDs, gradute students spend time in advanced courses and later on read lots of research papers and additional textbooks. This is an invaluable process of unsupervised learning (hence the first paragraph), and I think that getting a PhD also gives you the abilities and experience to autonomously go from zero knowledge in an area to contributing new knowledge by reading. Of course, studying at a university saves you a lot of time, because somebody else--the professor--who is already very knowledgeable in the target area has already broken down his knowledge in edible--and ideally pedagogically-sound--pieces ready to be sucked up. But if time is not of the essence, and since you are already a driven and focused person, you should be able to do it by yourself.
So this was (quite unexpectedly) rather long, a more detailed discussion and probably a very good book to read for anybody can be found in: Mortimer Jerome Adler's wonderful "How to read a book". A brief summary for tl;dr reasons: I love and totally support your reading advice, but since the PhD experience enables you to work your way through literature in unknown territory, it might very well be worth the effort.