I'm can't claim to have studied the relevant history in a lot of detail, but count me a skeptic of Landsman's claim. Let's take this little paper and the companion that it cites as a test case, which I hope we can all agree is "real physics". The authors are clearly well versed in the calculus of variations and the representation theory of Lie groups. Both of these subjects are heavily intertwined with functional analysis - functional analysis is even foundational for the former. Are we to believe that these physicists were entirely ignorant of the subject? Or is the argument that functional analysis only influenced them indirectly through its contact with those mathematical applications?

I think Landsman's argument makes an error common among pure mathematicians about how mathematics is actually applied to the sciences. We tend to think about theorems, because those are the main objects of study in our work, but for consumers of mathematics it is the definitions that are important. The role of theorems is to validate the correctness and importance of definitions, and sometimes provide tools for manipulating them. The definitions of functional analysis - (un)bounded linear operators, Hilbert spaces, states, and so on - appear all over the place in quantum mechanics. And many of the big open problems in theoretical physics call primarily for definitions rather than theorems: Is there a measure space on which path integrals make sense? What is the correct notion of Dirac operator on the loop space of a manifold? Is there a gauge theory which includes both gravity and the standard model? And so on.