It is probably dangerous to answer without reading Landsman’s whole paper (and the question seems likely to be closed as “opinion-based”), but I’ll record my first reaction as much like @lcv’s (*), namely, it sounds a little bit strawman-ish to separate the two and then pit one (FA) against the other (QM).
Footnotes (by Born) on pp. 583, 585 of the famous 1926 Dreimännerarbeit (see commented translation pp. 351, 352) immediately identified Hilbert’s work on linear operators as the correct framework for QM, and are a literal blueprint for its extension to unbounded operators by von Neumann. If this, and Weyl’s group reformulation of $[P,Q]=I$, are not contributions of functional analysis, then I don’t know what could be!
Consider also that, as far as could be determined under this question, the first two times our very phrase “linear algebra” (I don’t mean the thing) appears in the literature are (**)
In Hermann Weyl’s 1919 book on general relativity; the phrase didn’t catch on then.
In Hermann Weyl’s 1928 book on quantum mechanics: then it caught on.
So QM and FA both played a role in establishing the other as a field of study.
(*) Which I would qualify to: FA (some say linear algebra) “is essentially the same thing” as those parts of QM we understand.
(**) The phrase also appears, just once fleetingly, in Courant and Hilbert’s 1924 book, of which H. Hameka writes in his nice informed account, p. 11: “by a fortunate coincidence, linear algebra was the subject of the first chapter in the newly published book Methods of Mathematical Physics by Courant and Hilbert.” Born and Jordan cited it in 1925 (translation p. 279); like Landsman, I think the idea that physicists needed no one’s help came largely from Dirac’s failure to cite almost anyone.