I wouldn't know how to set up such a study, but I'd love to see how ideas that are simple, catchy, and wrong fare in the meme pool relative to others of slightly more complexity but also of more merit. On occasion something tremendously stupid just seems to stick. Usually the best place to see this is in the faux "debate" over global warming, but of late we've been hearing something especially silly about school choice programs such as vouchers and tuition tax credits.
The claim is that vouchers and the like are harmful to kids stuck in the legacy public schools (my words, not theirs!) because they result in a de-funding of the public schools. That's total nonsense: the amount allocated to the public schools is set independently. No state, including Arizona, has a law saying "We start with X, and then subtract what is spent on vouchers or the revenue lost to directed tax credits." The programs are decoupled. Vouchers and tax credits mean the state can spend less on public schools because fewer pupils will attend. They do not mean that the state will underfund the public schools. They are merely another expenditure. One could say, of course, that this means that less is left in the budget to spend on the legacy institutions, but to be honest about this one must also rail against roads, prisons, AHCCCS, and every other state expenditure as resulting in the public school kids being short-changed.
School choice does, of course, have public choice implications. I'll leave details to real economists but will sketch an argument for you. When some critical portion of children are being educated by private for-profit and nonprofit institutions in the marketplace, their parents, grandparents, relatives, neighbors, and communities in general will start treating keeping the legacy public schools afloat as being of dubious merit, and will punish legislators at the polls accordingly. "Why are you spending on that old-fashioned horse-and-buggy stuff? What a waste! Send the kids to private school like everyone else!" But when that point is reached, it'll be because we replaced the legacy system with something superior: a market in education, a system of schools in place of a school system, and private choice in place of democracy.