The Elections Clause does not vest exclusive and independent authority in state legislatures to set the rules regarding federal elections. Marbury v. Madison, 1 Cranch 137, famously proclaimed this
Court’s authority to invalidate laws that violate the Federal Constitution. But Marbury did not invent the concept of judicial review. State courts had already begun to impose restraints on state legislatures,
even before the Constitutional Convention, and the practice continued to mature during the founding era. James Madison extolled judicial review as one of the key virtues of a constitutional system, and the
concept of judicial review was so entrenched by the time the Court decided Marbury that Chief Justice Marshall referred to it as one of society’s “fundamental principles.” Id., at 177..

The Elections Clause does not carve out an exception to that fundamental principle. When state legislatures prescribe the rules concerning federal elections, they remain subject to the ordinary exercise of state judicial review.

(from syllabus of Moore v Harper)

[Thomas dissented, joined by Gorsuch, and Alito in part. Thomas’ main objection, however, is that the case should have been ruled moot on the basis of subsequent state actions. He also objects to the resolution but only in the sense that Thomas believes the issue of judicial review to be a non-sequitur. Notably, Alito did not join parts II or III of Thomas’ dissent. No one on the court seemed interested in establishing an “independent state legislature” theory when it comes to election law. Thomas only objected to widening the scope of federal judicial review. And this should come as no surprise, given the novelty of this argument in the wake of the 2020 election. — Ed]