Nowadays all the leading works on historical linguistics consider Proto-Slavic (you can put here anything you wish but I will talk about Slavonic studies) language as a fact (yes, there can be said something like, 'It is a hypothetical language,' but that is not serious). There are explanations of phonetical processes in this language, hypotheses on how the infinitive form appeared, etc.—so the scholars take the proto-language as something obvious.
Nevertheless, there were some linguists who disagreed with the reconstructing of proto-languages for the majority of IE (and other) languages. One of them, as I suggest, was Vittore Pisani. In his work 'L'etimologia: storia questioni metodo' he writes (here comes my translation from Russian):
Regarding 'the Germanic language', it is necessary to remember that it has never been the same as Classical Latin in which common to individual linguistic acts isoglottic lines were rather numerous and well known; 'the Germanic language' was sooner a branch of dialects (just like Vulgar Latin) with a far lesser number of isoglottic lines which are not regulated by a strict tradition.
But in the process of the forming of these [Proto-Slavic, Proto-Germanic, etc.] linguistic unities, every dialect or even every linguistic tradition could save or develop some ancient characteristics which probably had been its [of the dialect or of the linguistic tradition] isoglottic lines that are common for the dialects which then became a part of other unities. This shows how erroneous and fraught with delusions the idea of the evolution of language is...
Well, my question is how modern historical linguistics can answer to this 'accusation'? Why actually the model of proto-languages has been taken and is being developed now by modern scholars?
The problem you raise, that "Proto-Germanic" was a huge, blurry-edged mass of dialects and variations rather than a single standardized language, is a valid one. But that's a problem that also exists for modern languages. You can trace dialect continua from Rome to Lisbon, or from Prague to Vladivostok, without ever crossing a hard line of mutual unintelligibility. Why do we say that one village speaks Spanish and another speaks Portuguese, when they can understand each other just fine? Why do we draw these hard-edged lines between languages, when they don't reflect any objective fact?
The answer is, to put it simply: because the idea of a language is useful.
Even in places where no government is standardizing things, it's useful to set up an artificial model and say "this is General American English", or "this is Literary Lingála", or "this is Proto-Germanic".
Does anyone actually speak perfect General American English? Almost certainly not. But GA turns out to be a wonderfully useful model for studying English. For any given idiolect, we can figure out how it's the same and how it differs from our GA model (do you have the pin-pen merger?), and we can extrapolate from the model to idiolects with good enough accuracy to be useful (you don't speak perfect artificial GA, but I'm willing to wager you use DO-support).
Like anything in linguistics, languages and proto-languages are models. Arguably, anything except a native speaker's mental intuition is only a model. And like all models, they have flaws. But they've proven to be wonderfully useful, and so we continue to use them. For a final example, even if nobody speaks perfect Literary Lingála, this artificial model was used to develop the writing system for all Lingála idiolects. Its phonology is only a model of the way any given person in the Congo actually speaks, and it's simplified, missing all sorts of phonetic details and allophones. But it's "good enough" to represent the language unambiguously, and that's what matters.
The concept of a proto-language stands and falls with a tree model of language evolution. While the tree model of language evolution can be successfully applied to many language families leading to widely accepted reconstructed proto-languages, it is not the only possible model of language evolution.
In a situation where there is a polycentric dialect continuum with innovations spreading from different places, a wave model may be more appropriate to describe language evolution. Pisani is a well-known proponent of a wave model of language evolution so he denies the reconstructability of a unique proto-language.
You have to keep in mind that the comparative method means several steps:
step1 is collecting comparanda: for example, Latin pater, mater, frater vs English father, mother, brother.
step2 is creating a formal equation, then proposing a pre-form that synthesizes the equation: here for example *pater, ma:ter, bhra:ter (at a naive level)
step3 is projecting the equations *pater, ma:ter, bhra:ter into the past and postulating that these equations are words in the proto-language.
Nowadays, people are so routinely applying the comparative method that they jump to the conclusion, without explicit reference to step1 and step2
Basically, a proto-language is a set of formal equations that synthesize the similarity between sets of comparanda and are projected into the past.
First off: One should not make the mistake and equate ancient languages with today's. There is an important difference between having a) the bigger half of the population live in remote areas with changing rulers every full moon, with a tremendous analphabetic rate; and b) having a reporter every evening read the prestige dialect or standard form over the air in a stable country. Anyhow, remote areas still exist, so comparison in situ is surely done.
The struggle for a unified language is very real today for Serbo-Croation for example--or not, depending on which side of the fence you sit on. However, from a historic perspective, for the most part, the winner will leave the majority contribution to the imprint of the language.
The truth is, that language consists of idiolects per person that can change somewhat over time. That's of course beyond impossible to work with, though dialect research is a thing, of course, so researchers necessarily have to simplify. This is not quite so bad under the assumption that a native speaker will operate from a rather inert core--that's why L2 accquisition is so diffficult passed a young age (and that's why writing has to simplify. Therefore the trunk of a languages development is supposed to be reconstructable quite easily, even if taking stats for only one semester just barely passing. The branches, fruits and everything else is rather difficult;
The discovery of a common Indo European family is barely two centuries old. The field always adds disclaimers, for good reasons. It's work in progress, few people have the prowess to engage, little is certain. Traditional etymology is littered with false etymologies that have become part of the language and need to be separated, religious or nationalistic nutters are trying to contribute, scientific rigor and interconnected exchange are kind of a new thing, too. The biggest showstopper is always the homeland questions, that requires experties outside of the field, and evidence that is burried or lost. That means, I fully agree with the sentiment of the question.
In order to understand the reconstructions, one has to learn them like any language, that makes it feel very real. However, there is no one proto language; The ideas differ like idiolects do, except for a common core. The core words of an idiolect, like most any words as well as the reconstructions are symbolic. If concensus emerges from the words, then they have become defined. For a research language, that means these words reflect sensible relations. We can call that a language, because the concensus on the definition of a language is rather weak! Just ask a linguist what a word is and you will get many opinions.
The assumption that there was at least one cohesive language is simply necessary. There can't have been zero, after all. Whether there were more is an interesting question, and slight differences at common joints may be worthwhile to consider. However, potato, potatoe, is just seen as an aspect of realization. Alternative spellings, if one can call it that, are given due to different results. Focusing on just one restrictive theory can be a real problem, but it's also the goal. That's the real problem.
Language needs to be locally and temporally bound, thus a working theory needs to try to stay within the bounds. Nobody makes strongly specific claims to the extent of the boundaries. In mathematical topology, disjoint areas can still be treated as a unique set. In that sense, the boundaries could be quite complex. Nobody is really talking about a single point, it's just an abstraction.
Graded dialect continuums are not unheared of in the field (e.g. for PIE in Gramkrelidze & Ivanov; I have no idea about Slavic specifically). The possibillity of Substrate Languages has caused various speculations, too.