As adminitude noted, this is not typically the way it works. Many of the anti-atheistic arguments developed within specific social and religious contexts in which the local orthodoxy could be confidently contrasted with atheism as the only two reasonably possible options. Accordingly, the rationalist arguments you reference are usually about the existence of God in general, and defend most centrally a conception of God sometimes called "God of the Philosophers," the abstract unity of all perfections, rather than a personified conception of God. There are other kinds of arguments, which are not necessarily the same (or rather, necessarily not the same) for the reasons to prefer one religion over another, first given the existence of God, and still other kinds of arguments around the specific points of doctrine that divide one sect from another.
As the world globalizes, the problem of defending a specific religion simultaneously against atheism and against other religions becomes more acute, and one might expect new arguments to emerge with this as a goal. The only one I'm personally familiar with, however, is Kierkegaard's argument for Christianity, which is actually anti-rationalist, and centers around the paradoxical concept that it is the very absurdity of the Christian narrative that certifies its validity. (I believe CS Lewis also tackles this issue head on, but I don't have any specific citations available.)
As far as why people don't just advocate a generalized non-specified theism, some do, but the historical record seems to indicate that people in general tend to find this approach considerably less compelling. Speaking personally, as a person of faith, I would say that the abstract arguments are comforting and fulfilling at an abstract and intellectual level, but that for the challenges of everyday living, you need a faith community, and all the specifics that go along with that. One might not be able to defend that second choice, however, with the same formal rigor as the first.