Is there a more modern text that someone could suggest that covers roughly the same material as does Whitney's Geometric Integration Theory? There are a number of topics in this text that I haven't seen treated elsewhere.
I don't think there is another book which covers all the same topics. I am only really familiar with a small portion of the book concerning the approximation of smooth manifolds with simplicial complexes. As far as I am aware, there is no other book that compares as a resource on this topic. Munkres's 'Elementary differential topology' is very good for this also, but it is hardly a more modern treatment.
I think the book by Bott and Tu "Differential forms in algebraic topology" would be an example of a more modern treatment of much of the material in the first half of Whitney's book.
I don't think there is a modern "branch" of mathematics into which Whitney's book would comfortably sit. There is geometric measure theory, and there is a book by Krantz that even has the same title as Whitney's book, which I guess you've come across if you googled that, but this has quite a different flavour from Whitney's book. It is not the same topic really, even if the same words do well to describe both.