They, the politicians, would become irrelevant and that is anathema to a politician especially a democrat. For this reason no democrat or republican politician for that matter would ever want to cede control of people and their money. Citizens who are self sufficient and who know what they want for their kids have no need for meddling, destructive and expensive politicians. As a Libertarian I truly believe in choice and the empowerment of individuals. The country was founded on the principle of Liberty, that is the individual decides what is best for him and how to spend his life and money in that pursuit of happiness as Mr. Jefferson put it in the Declaration of Independence.
School vouchers are a step in the direction of that choice.
Libertarians would prefer a system of tax credits where people could take a dollar for dollar tax credit and pay for their own children's schooling. Where there are needs for poor people and their children the credit could be taken by other individuals or companies that have no kids but would like to see all kids educated. This way everyone could chose the kind of school they want, secular, parochial, Christian or any other. They could choose schools with uniforms, prayer or even single sex schools. They could choose schools that provide other or even special needs that the parent determines is important.