Schools and most welfare programs are state functions, the state has every right to decide who gets in.
Plyler v. Doe, 457 U.S. 202 (1982):
The Court held that A Texas law which denied the benefits of free public education to illegal aliens violated the Equal Protection Clause of the Fourteenth Amendment, stating that discrimination on the basis of immigration status did not further a substantial state interest.
A ridiculous decision, obviously, but that's how we got to where we are today. Justice Brennan's footnote #10 gave us birthright citizenship for illegals. It could probably be eliminated by statute.
That is debatable. Not that I disagree with you by and large but with so much Federal involvement in those areas it is not as simple as asserting they are solely State programs.
Immigration laws are under the Federal government. The problem and the reason for State government response is the failure of the Federal government to fullfill its constitutional duties regarding immigration.
If the FedGov is going to attack you at ever turn, it's probably not the right time to try these maneuvers.
Perhaps with the *next* admin?