Why not just create specific religious schools like the Catholics do?
Yep. Dred Scott held that blacks could legally be classified as property unless freed, and that non-slave states had an obligation to return them to their owners. Plessy v. Fergusen held that they could be kept separate from whites, provided that they were afforded all the rights, privileges, and immunities of US Citizens. There are cases in line with Plessy that allowed certain colleges to put up sheets in between sections of students, one section for whites and one for blacks. Sometimes the Court gets it wrong; that's why, on rare occasions, they reverse past precedent (as Brown v. Board of Education I and II did). If that doesn't work, we have an Amendment process (the 13th Amendment effectively reversed Dred Scott).
Still a pretty half-baked attempt to discredit them, however, considering that they're the ones who get to interpret the law and settle Constitutional disputes. Until the current (and long-standing) trend of non-interference with religion ("shall neither advance nor inhibit") is reversed, however, the law of the land means that the Government has very little business with religion, whether that be involving intelligent design in schools or creating permits to allow for proselytizing (a CT case).
Well lets hope Missouri is smart enough not to fall into this pit.
Point 1 is, well pointless since they have that already.
Point 2 - Not sure how legal it would be for it to be on site during school hours in the US. It will blow up the moment the "wrong" religion insists on using this right anyway.
Point 3 - Just a hook to catch a few more votes, who would vote against the BoR?