The title and the question ask something different. "Is X sufficient" versus "is there a single instance of Y". Some examples of Y have been proferred.
A very mathematical approach, find a single counter example. However the question is sufficient is actually a value question. One does not have to (I would argue should not) accept the idea that one needs to know everything.
Looking at the title, a little more holistically, I would actually answer "pretty much so" to the question of if R integration is sufficient.
Let's think about it practically: are there any homework problems in the following courses, that you need L integration to solve?
Undergrad: -calc-based intro physics? No. -junior year E&M? No. -junior year mechanics? No. -"modern physics"? No. -quantum theory? No. -thermal physics? No. -undergrad electives in optics, quantum optics, acoustics, nuclear, sonar, solid state, etc. No!
Grad school: -Jackson E&M? No. -Mechanics? No. -Quantum? No. -99.44% of all the other courses? No.
So...you can learn a heck of a lot of physics, get a degree, and have a lot of commercial value without learning L integration. Note, how this is different from other concepts that you do need at various stages (integration by parts, PDEs, etc.) Compare how dramatically different the use of R integration is with the abstract examples offered up here.
P.s. I have done a lot of USE of probability and statistics in physics, chemistry, engineering, business, and psychology. And I did not need L integration (I don't know it!)