R. Rankin's 1939 paper giving a non-trivial estimate on Ramanujan's $\tau$ function used the "real-analytic Eisenstein series" for $SL_2(\mathbb Z)$, at least. Selberg's related paper just-slightly later seemed to express the same awareness for such cases, as opposed to the more general situations treated in the 1950's. In some informal remarks I saw elsewhere, Rankin credited his advisor, Ingham, with knowing how to meromorphically continue such Eisenstein series.
The idea for congruence subgroups of $SL_2(\mathbb Z)$ was recapitulated (at least) in one of R. Godement's articles in the 1965/66 Boulder Conference proceedings (AMS Proc. Symp. IX), which was where I saw it first.
The appendix in Langlands' SLN 544 in which he stitches together several $GL_2$ cases to get minimal-parabolic Eisenstein series for $GL_n$ and $Sp_n$ certainly presumes that the $GL_2$ case is well-known. Apparently these notes were written up by about 1967 even though they were not public until 1976.
My own (then-naive) impression by the mid-1970s was that the meromorphic continuation of $GL_2$ things was a cliche. Selberg's ideas from the 1950's seemed to suffice for rank-one situations without subtle data on the (possibly non-abelian) Levi components. Lectures of G. Shimura at Princeton in the mid-1970's proved things, including meromorphic continuation of Eisenstein series, by looking at Fourier expansions of all sorts of $GL_2$ Eisenstein series, related to applications of Maass-Shimura operators and "nearly holomorphic" automorphic forms on $GL_2$ over totally real fields.
(So my own 1989 book on Hilbert modular forms gave such an argument over totally real fields, maybe mentioning Rankin and Selberg from 1939...)
So, although it might have been reasonable for (e.g.) Iwaniec to mention Kubota, I think it is not accurate to worry that a deserved citation was inappropriately denied. Rather, already by the time of Kubota's book those ideas were perhaps considered "general knowledge", so by the 1990's (and Iwaniec' book) the ideas would have been even more "classical".