This is what they call a "good question", as the expression goes. I and my collaborators had it in the mid 1990s, and about a 3rd of the audience in my colloquia ask it to themselves and me, rarely getting satisfactory answers.
- The short answer must be that there are no "killer apps" of it, namely problems which this formulation solves dramatically better than the other two formulations.
The "gold medal" one, operators in Hilbert space, is what is normally used in atomic, molecular, solid state, and subnuclear physics, by dint of practicality.
The "silver medal" one, functional integrals, languished likewise for two dozen years, until it was used to quantize gauge theories (ghosts), organize their Green's functions, and ultimately have them effectively solved by lattice computer simulations.
But the "bronze medal" one you are asking about has merely helped several researchers, including ourselves, to think more clearly about problems they solved; it cannot be argued, in retrospect, that the solution would have been impractical in Hilbert space or path integrals, only harder. So this formulation, so far, is mostly good for providing intuition on subtle problems, including visualization of the classical limit and killing rampant misconceptions on quantization schemes. Its tasteful math is also useful in Lie Algebras (* product, Moyal Brackets, large N limit), but the mainstream regards such things as mathematical physics diversions.
I know some of the history of the subject; here is our review of it. We conjecture there that, in the late 40s, the formulation was eclipsed by path integrals, as well as the formulation of QED by Feynman (who, like Dirac, was intensely interested in it), and which "stole the show". By looking at the literature over the years, I've confirmed the impression of H Corben (quoted in Moyal's biography by his widow, Ann Moyal, now also deceased) that its misapplication in the quantization problem by wrong-minded adherents of Weyl's paradigmatic quantization dream damaged the formulation significantly. Frustratingly, it took the smoke to clear about 50 years, if that.
A corollary to this is the ignorance of Groenewold's * product and its significance, and Groenewold's epochal thesis/paper at large. Even today, see below, it looks subdominant in citations to Wigner's and Moyal's, even though it is conceptually co-equal, if not over-arching. And to think that Hip worked so hard to have it published in the Physical Review, and failed.
However, let me flesh out your picture by including three citation ("knowledge") graphs versus time, for the three seminal papers of the subject. They all exhibit the same peculiar pattern. I'll use the plots of INSPIRE, which focusses on HEP, but has ready plots, which I (perhaps unjustifiably) don't know how to access in Google Scholar, far more complete. The sampling by INSPIRE, however, follows the time development of the larger GS set.
Wigner, E., 1932: On the Quantum Correction For Thermodynamic Equilibrium, Phy Rev 40(5), p.749; Wigner's 1932 paper has 11367 cites in GS, and 1414 in ISPIRE, evolving as 
Note the pitiful performance for half a century.
Groenewold, H.J., 1946: On the principles of elementary quantum mechanics, Physica 12(7), pp.405-460; Groenewold's 1946 thesis has 1528 in GS, and 524 in INSPIRE, and evolves as 
, again bizarrely neglected for half a century.
Moyal, J.E., 1949: Quantum mechanics as a statistical theory, Mathematical Proceedings of the Cambridge Philosophical Society 45 (1) pp. 99-124; Moyal's 1949 (well, really 1941 or so...) paper has 3868 in GS, and 1078 in INSPIRE, 
. The scale, for whatever reason, only goes back to 1953, but note in its midpoint year, 1987, it only has 13 citations, and the next year, 1988 only 4, the dip you can identify in the middle. Again, it is languishing for half a century.
The awareness & appeal of that field only started crawling up in the 1990s, and then got going. I suspect a professional historian of science should have insights on that shift, but my gut feeling is that it's just generational: an entire generation of physicists with the wrong (paradigm) focus ("what is the correct quantization path from classical mechanics?") had to die, before the modern viewpoint ("what more can we learn about QM from this equivalent formulation?") could thrive. Just a thought.
