"I've come to the conclusion that the logic becomes "well if I can't have it, no-one will"."I don't think the poor who consistently vote Republican really care about the economics of social services. That's too removed and abstract a concept for them (ironic, given that it affects them directly). Instead, they're voting emotionally: on social values, and on the politics of identity and nationalism. These abstract concepts are very "real" to a lot of Americans, because they're the things people cling to and hold fast when everything else is taken from them.
Bear in mind that a lot of these people have seen social values work for them amidst their own circumstances. Those among them who work their asses off, keep out of trouble, go to church, stay clean, and don't overspend will do markedly better than their peers who slack off, abuse government assistance, and engage in reckless behavior. Meanwhile, a preponderance of military personnel are drawn from the families of the working poor -- so these folks eat, breathe, and sleep patriotism. They love America, even if the version of "America" they love is largely a myth.
Republicans have become masters of speaking the language of identity politics and values. They understand what tune these guys want to dance to, and they play it flawlessly. Meanwhile, Democrats are tonedeaf. Nevermind that Democrats may actually have the interests of the little guy in mind; they don't speak the little guy's language. And that makes all the difference.
Case in point: the Democrats consistently talk about "the middle class," and sometimes "the working class" or "the poor." Meanwhile, Republicans always talk about "Americans." They rarely even speak of classes or social divides. It's a brilliant rhetorical gesture, because it recognizes that the working classes don't enjoy hearing about how shitty their lives are. They don't like the labels. If you ask them, many of them won't even agree that they're "working class" or "poor." They're in denial, and it's a defiant denial. They're too proud to face reality, and when Dems constantly throw the reality of their misery in their faces, they get angry and resentful.