Don't think it has much to do with the notation. Take the integral sign for example. What are we talking about Riemann, Riemann–Stieltjes, Lebesgue, some weird stochastic integral? It's the same symbol. You won't "understand" it better by using a different symbol. Even the Riemann integral, to rigorously "understand" it requires lots of concepts from analysis like least upper bounds (infimum, supremum), partitions, etc. The notation is hardly the stumbling block here, there's no way of getting around understanding the details if you really truly want to "get" it. But again, in practise, when you're doing integration by hand, you've learned a few tricks that work and you hardly ever think about it at the level of analysis.