The actual "physical" roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians' attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.

I was talking about the applications of functional analysis to understanding differential equations, which are (as I understand it) the actual point of functional analysis.

This sounds like violent agreement to me.
Your disagreement is the utility of the application of functional analysis to differential equations. Is it a practical problem to know when Dirichlet's principle applies? Or, if you insist that functional analysis dates from Leray, I am told that physicists do not care about the mathematical problem of whether the Navier-Stokes equation has smooth solutions--water flows, and that is good enough for them.