I've only studied measure theory in a probability context. Can you talk a little bit about why the Radon measure isn't sufficient there, and what it is useful for outside of probability?
Bourbaki's shortcut to Radon measures is very elegant but it's noteworthy that unlike many other Bourbaki innovations I don't think it was picked up by other textbook authors. Already at that point there was a mathematical consensus that measure theory was a valuable part of the foundations of modern mathematics and shouldn't be eliminated or minimized.
Outside probability theory, measure theory is primarily used as a foundation for integration ("expectation"). There are also more specialist subjects like geometric measure theory; there's an excellent introductory textbook called Measure Theory and Fine Properties of Functions, and if you look at its table of contents you can get an idea of the breadth of topics.