Actually, the Supreme Court WILL entertain arguments that were not advanced below - as long as the actual causes of action that are buttressed by those arguments, are not new. So, if you brought a First Amendment cause of action, and supported it below with the argument that government unconstitutionally prohibited you from saying something, you could add, at the Supreme Court, an argument that government also FORCED you to say something - which is also a First Amendment violation. "Once a federal claim is properly presented, a party can make any argument in support of that claim; parties are not limited to the precise arguments they made below." Yee v. Escondido, 503 U.S. 519, 534 (1992); see also Dewey v. Des Moines, 173 U.S. 193(1899).
Problem with that principle in the Schindler case, is that the Supreme Court, I believe, is reviewing the lower court ruling (denial of prelim injunction) for abuse of discretion, so such a review might have to confine itself to the arguments that were presented to the court below - in order to judge whether its rejection of them constituted abuse of discretion. On the other hand, given the gravity of the life-and-death issue, the court might employ a more exacting analysis, asking if additional arguments for, say, a Due Process violation COULD have been propounded by the court itself but weren't (as a court might have been expected to do if the plaintiff is a non-lawyer representing himself).
I'm wondering if either of you know why the Schindlers have not had their causes of action tried as of yet. Why didn't they push for trial when injunctive relief was denied? Were the original pleadings geared only toward injunctive relief without including a plea for declaratory judgment and damages?