[WS:] I think this confuses ownership and purpose. Institutions owned by religious organizations can receive public funding if they serve public purpose. Hospitals are a good example - religiously owned hospitals (e.g. Catholic or Adventist) receive Medicare or Medicaid payments, which are equivalent to vouchers, without violating the separation of religion and state. The same can in principle apply to schools.
However, public funding cannot be used for religious purposes, such as worship or proselytizing. But this is also true for political activities as well. All tax exempt organizations under the IRS section 501c3 cannot engage in political activities, defined as promoting specific legislation or a specific political candidate, on the pain of losing their tax exempt status (it does not happens often, but the IRS does revoke the tax exempt status from time to time.)
The way the nonprofits resolve it is that they spin off organizations registered under a different law (IRS Code Section 501c4) that does not grant wide tax exemptions but permits certain political activities. This way, they can have the cake (receive tax exemptions) and eat it (funnel it to their 501c4 spinoffs).
In the US, the problem is not with schools per se, but with the population attending them. Public schools suffer from two related problems: (1) they have to serve everyone, including subpopulations that due to an array of social problems are not ready to receive education, and (2) they are highly politicized.
The attraction of non-public schools (both private and charter) is that they seem to offer the escape from these two perils. They seem to remove students from the social problems brought to public schools to be addressed there and they seem to offer learning environment free from political meddling (such as standardized curriculum or standardized testing.) That is their main appeal to parents who sufficiently care about their children's education.
I am a big supporter of public education, but I also think that the voucher system for non-public schools is not necessarily detrimental to it. First, it can allow opting out of the "Drang nach universitaet" - with standardized curriculum and standardized testing that the public schools are currently forced to enforce under severe penalties. Furthermore, there is no law that prohibits unionization of non-public schools. Although I am not aware of any serious union organizing of non-public school teachers, this seems to be more a problem of union complacency than any actual obstacles to unionization.
-- Wojtek
"An anarchist is a neoliberal without money."