This order of the Supreme Court means only that if a case arises from a particular judicial circuit, then the justice so assigned shall receive it first. A judicial circuit in this context encompasses the circuit court of appeals, the federal district courts, bankruptcy courts, state courts, and other tribunals from which the Supreme Court may hear appeals. I am not sure of the pertinence in original-jurisdiction cases. For this order to have relevance, someone must bring a lawsuit to a lower court, which then must rule on that lawsuit; a party to the lawsuit then can appeal to the Supreme Court, at which time the named justice will conduct an initial review of the case and refer it to the full court. The named justice in some cases upon request can issue an emergency order or stay while seeking the input of the other justices.
Our Twitter commentator apparently thinks that if an election-related lawsuit reaches the Supreme Court, then it will find a relatively slightly more favorable audience with the assigned justice. This favorable treatment may result in a marginally greater chance of the Supreme Court accepting such an appeal. It perhaps may increase the chances marginally of a temporary emergency stay, but the justice for the circuit probably will not act without the concurrence of other justices on the court in a dramatic political case.
i am under the uninformed impression that the USSC has both reach down and reach up emergency procedures for time urgent cases. (someone somewhere surely can confirm or deny...)