My hunch is (like always) there are people in both groups, but what was the original purpose of the theories? Are they mainly linguistic tools for describing various syntactic phenomena, rather than trying to explain cognitive language structure?
The question in the title seems rather different from the two questions in the body of the text, and these two in turn are not synonymous either (the original purpose of a tool may be quite different from what a tool is mainly use for).
At any rate, my answer to the question in the title is that they are legitimate theories (because an illegitimate theory is a quite rare beast, it seems to me). The original purpose of any given scientific concept is always hard to ascertain, though in that case one can confidently say that phrase structure rules were not originally introduced to describe cognitive structure, for the simple reason that their popularization in linguistics occurred before the idea that there might exists cognitive structure involved in language became largely accepted (and in fact, the development of phrase structure triggered the cognitive reevaluation, not the other way round). As far as I know, the idea that phrase structure grammars could be an accurate model of cognitive capability appeared for the first time in Robert Lees' review of Syntactic Structures in 1957, whereas dependency relations were explicitly introduced at the latest in the XIXth century and represent a linguistic tradition coming back much further in time (see Thomas Gross comment).
Regarding the last question,it seems to me that our knowledge of syntactic phenomena is still thin enough so that there is no meaningful distinction between accurate description and theoretical modeling of the cognitive process involved. That said, it is clearly the case that certain strand of phrase structure grammar (minimalism for instance) are prone to posit and work with linguistic concepts which have no immediately accessible linguistic incarnation (for instance unpronounced abstract nodes in trees or abstract Case) whereas other strands dismiss the process as unscientific (angels on pinheads in the words of frequent contributor @jlawler).
Lucien Tesnière, the father of modern dependency-based theories of syntax and grammar (i.e. dependency grammars, DGs), famously wrote the following in Chapter 1 (paragraph 5ff.) of his main oeuvre Éléments de syntaxe structurale (1959):
"a sentence of the type Alfred speaks is not composed of just the two elements, Alfred and speaks, but rather of three elements, the first being Alfred, the second speaks, and the third the connection that unites them – without which there would be no sentence. To say that a sentence of the type Alfred speaks consists of only two elements is to analyze it in a superficial manner, purely morphologically, while neglecting the essential aspect that is the syntactic link. [...] The connection is indispensable for the expression of thought. Without the connection, we would not be in a position to express a single continuous thought and we would only be capable of producing a succession of isolated images and ideas, with nothing linking them together." [Translated from the French]
What Tesnière calls a "connection" here is what most modern DGs call a "dependency". The passage demonstrates that Tesnière had a mentalist understanding of dependencies. One could call it a cognitive perspective. He viewed dependency structures as cognitive structures.
I disagree with the sentiment expressed in Olivier's answer above ("an illegitimate theory is a quite rare beast"), and implied in a sense in Jlawler's comment above ("A large number of serious syntacticians believe that one needs both dependency and PS relations"). An extreme version of that sentiment states that all theories are legitimate theories. The one theory, no matter its nature, is every bit as good as the next. That is a kum-ba-yah view of science that I reject, i.e. "your science is as good as my science; let's all be happy", and if we take it seriously, it would actually mean the death of scientific progress.
Science -- in this case theories of syntax and grammar -- are not all equally good. Some are better than others, and if we are interested in scientific progress, we should constantly be comparing the predictions that theories make. The theories that are making accurate predictions with the least effort are the better ones.
I think that a comprehensive theory of syntax needs both dependency and constituency, but not in the way that Jlawler's comment seems to imply. Constituency is needed only for coordination; it is not needed for subordination. Coordination aside, dependency beats constituency hands down. Dependency makes more accurate predictions than constituency with much less effort. It is in general the better approach to the syntax of natural languages. But that is of course just my view; each linguist has to decide this issue for themselves.