It's actually not clear that the establishment clause in the first amendment prevents federal funds from being given to or benefiting religious schools. A complete ban of government doing anything regarding religion doesn't actually exist, only separate non binding writings endorse the concept. There have been rulings that allowed funding of religious schools. The current precedent is the Lemon Test which has three parts that must be true.
- The statute must have a secular legislative purpose.
- The principal or primary effect of the statute must not advance nor inhibit religion.
- The statute must not result in an "excessive government entanglement" with religion.
There are also the coercion and endorsement tests that prevent the government from appearing to coerce citizens to support a religion or provide an endorsement of a religion. Since this amendment has been Incorporated as you mentioned it applies to all levels of government, which includes local.
A government could never "run" a religious school, as that would most likely failed the third part of the Lemon test. A government could in theory provide nearly all funding to allow a religious school to operate provided the method of funding applied to more than just religious schools, was focused on non religious education. It would also likely require that the school didn't restrict admissions based on religion and could show that religious teaching was funded by non government funds.
A voucher system would likely allow for religious schools to get government funding, as it satisfies most the above scenario and the choice is placed in the student's/parent's so the government s not endorsing anything. This also happens to be a major criticism from opponents of voucher systems in the U.S. as it would take money from secular public schools.