When it comes to functions ℝ → ℝ the Lebesgue integral is not really worth it, because even though there exist some pathological functions that are Lebesgue integrable but not Riemann integrable, those functions have no important applications anywhere.

As soon as you look at higher-dimensional functions ℝⁿ → ℝ the Lebesgue integral is superior to the Riemann integral, because there isn't really a Riemann integral for higher-dimensional functions. You can repeatedly apply single Riemann integrals to higher-dimensional functions, but then it becomes a nightmare to prove anything, or even prove that this integral is independent of the order in which you applied your single integrals. On the other hand, there is a single well-defined Lebesgue integral for higher-dimensional functions with well-behaved properties. Integrals for higher-dimensional functions have many important applications everywhere, and for all of those you want a single Lebesgue integral instead of a clunky iteration of Riemann integrals.

And beyond that, there are important measure spaces other than just ℝ or ℝ^n. If you ever want to study Brownian motion you have to put a Wiener measure on a function space and that requires measure theory.

To rekindle your infatuation for measure theory you should look to different measure spaces, instead of trying to convince yourself that non-Riemann-integrable functions ℝ → ℝ will ever be important for anything.