A part of the question thusfar unattended is whether the concept of phoneme has always been contentious. It has.
The position defended in Sapir's (1925, 1933) classic papers, that the phoneme is not just a unit of analysis, but has some psychological status, has always had its critics. There are two major facets of this claim that were challenged: that mental processes of language users involved a segment-sized atomic unit, and that this segment-sized unit was a primitive, called the phoneme.
Sapir's position is challenged by Twaddell (1935: sec.2), who is suspicious of the whole enterprise of inferring mental entities. A fine quotation from that work is given here:
Such a [mentalist] definition [of the phoneme] is invalid because (1)
we have no right to guess about the linguistic workings of an
inaccessible 'mind', and (2) we can secure no advantage from such
guesses. The linguistic processes of the 'mind' as such are quite
simply unobservable; and introspection about linguistic processes is
notoriously a fire in a wooden stove. Our only information about the
'mind' is derived from the behavior of the individual whom it
inhabits. To interpret that behavior in terms of 'mind' is to commit
the logical fallacy of 'explaining' a fact of unknown cause by giving
that unknown cause a name, and then citing the named x as the cause
of the fact. 'Mind' is indeed a summation of such x's, unknown
causes of human behavior.
A similar criticism, this time by a British phonetician, is found in Butlin (1937), who maintains (reflecting the then popular instrumentalist philosophy of science) that the phoneme is a "convenient fiction." Abercrombie (1991: 27ff.) [whose ideas were developed much earlier than the citation date suggests] cites Butlin's paper, disputing the status of segment-sized atomic units in language.
Eventual attacks on the phoneme from the side of generative linguists are anticipated by Martinet (1949), who argues that while most phonological concepts should be regarded as fictions, it is necessary to select at least one having status in reality, so that the theory could be grounded in facts describable in the languages of other branches of science. The concept he selects as "...la seule unité pour laquelle nous postulions une existence réele..." (1949: 46) [the only unit for which we postulate actual existence] is the feature rather than the phoneme. Martinet, however, appears to favor a non-mentalist conception of the feature.
Beginning with Halle (1954), American phonologists in what would become the generative tradition started to argue that the segment has psychologically real status, but that the phoneme is epiphenomenal, its observed properties actually being due to featural processes.
A very nice historical introduction to phonology prior to the 1970's is Fischer-Jørgensen (1975).