Edit (in response to Hugo Chapdelaine's clarifying comments): Selberg's spectral approach seems/seemed limited, if one insists on a full spectral decomposition, to situations where no cuspidal-data or other subtleties enter. My colleague D. Hejhal confirmed my suspicion (from Selberg's unpublished papers) that Selberg had not considered (to whatever purpose) cuspidal-data Eisenstein series, nor corresponding parts of a spectral decomposition for higher-rank groups. For that matter, so far as I can tell Harish-Chandra's relatively early results on automorphic forms on general semi-simple groups (e.g., 1959) were aimed primarily at the cuspidal part of the spectrum. Especially the residual part of the spectrum, not in the functional analysis sense but in the sense of being a residue of Eisenstein series, is the most ... obscure... part of Langlands' 544.
Yes, the Lax-Phillips 1976 argument that not only do spaces of cuspforms discretely decompose (e.g., for a relevant Laplacian), but even "pseudocuspforms", was a striking point, although I think their literal result was considerably obscured by its context, and a certain unclearness about how much was a physically-based heuristic and how much was really proven/provable. Nevertheless, yes, Colin de Verdiere's 1981 "new proof" of meromorphic continuation of Eisenstein series showed the peculiar potential in that fact, even though (to my recollection) many people were confused about the precise nature of "Friedrichs extensions", and how it could be that certain truncated Eisenstein series (palpably not smooth) could become eigenfunctions for some variant of an elliptic operator. (I recall wrangling between P. Cohen and Selberg at Stanford over such points c. 1980, and at the time I was certainly baffled.)
My perception at the time of the reception of Colin de Verdiere's result was that, mostly, since it reproved a true, known result, was just an "announcement" (rather than detailed paper), and used a tricky feature of unbounded self-adjoint operators, people did not engage with it as much as they might have. Nevertheless, W. Mueller showed in the 1990s that this approach certainly immediately applies to rank-one cases, and H. Jacquet at least sufficiently assured C. Moeglin and J.-L. Waldspurger that the analogous argument would succeed generally, so that they outline a similar argument in their book.
Of course, in all these cases the crux of the matter is that some artfully arranged operator has compact resolvent, therefore discrete spectrum, therefore various things have meromorphic continuations... Thus, in contrast to Roelcke's meromorphic continuation up to $\Re(s)=1/2$, which follows from very general spectral considerations, getting beyond that line when it parametrizes the continuous spectrum is non-trivial.
Yes, in fact, by now, I like the Colin de Verdiere argument better than other forms, since it gives more meaning to poles. In slight contrast, the J. Bernstein reformulation of Selberg's "third" proof (the latter from Hejhal's trace formula volumes) is a general-enough idea that, ironically, it seems not to obviously give sharp information about the case-at-hand, namely, automorphic forms.
One should also mention some experiments done by A. Venkov not only about spectral expansions for automorphic forms, in the usual sense, but also some Schroedinger operators or Dirac operators for automorphic forms...
I think the most serious subtlety is still correct identification of residual (=residue of Eisenstein series) spectrum. Harish-Chandra's SLN 62 from 1968 explicitly doesn't attempt this, and somewhere there's a quote from Langlands that he doesn't remember how that part of his SLN 544 went. For $GL_n$, Jacquet's conjecture from about 1983 (Maryland conference) proven by Moeglin-Waldspurger (1989) is the only (to my knowledge, at this date) systematic result for any family of groups. Of course, various methods can prove that some Eisenstein series have no poles in various cones/half-planes, but that's not the question.
For that matter, residues of maximal-proper-parabolic Eisenstein series are relatively tractable, via Maass-Selberg relations and their implications (as Borel already did in his little $SL_2(\mathbb R)$ book), but iterated residues entail complications that are not trivially dispatched.
