It's been forever since I looked at this stuff, sorry. But I think Bourbaki's shortcut to integration theory via continuous linear functionals only works in locally compact spaces. It doesn't let you construct the Wiener measure on a path space corresponding to Brownian motion or other continuous-time stochastic processes. And the example I gave of the conceptual, information-theoretic role played by sigma algebras in stochastic processes shows that sidestepping them is the wrong move for probability theory, even if they weren't used for some of the advanced technical constructions like Wiener measure.

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.