The Electoral College vs the Equal Protection Clause

Right, but the relevant question is not who implemented the winner-take-all system but whether that system violated equal protection, and that the assertion that it does is not found in Williams. There, the state acted invidiously to favor particular associational groups, whereas in setting up a winner-take-all system, the state is acting neutrally as among the parties. Nothing that the state is doing in implementing winner-take-all is inhibiting the ability of the parties to compete in the election. That’s what Williams appears to stand for.

Certainly, there may be a legal argument that the constitution requires a certain allocation of the results of the election. The analysis linked to by the OP, however, just does not make that case.

Yes. That’s what “equal protection” means.