Aren't the states more or less free to conduct elections however they decide? (including for federal positions).
Sure, as we have discussed here before.
From a skim of TX's court documents, they seem to be arguing that the COTUS says that state
legislatures can set the rules on how an election is run in that state, pretty much how they want. TX is claiming that the defendant state's Executive and Judicial branches changed the rules without legislative action, thereby not providing the election that had been decided on by the party empowered to decide such things. As such, the elections themselves don't conform to the COTUS (i.e. an election run as the state legislature decided to run it).
TX isn't, in fact, asking for the elections to be thrown out. They are requesting that the SCOTUS kick the decision of who the state's electors will be back to the State Legislature of the various defendant states, to decide how they wish. That way the picking of the states electors will have been done by the state legislature, as demanded by the electors clause of the COTUS.
For example: the PA legislature laid specific rules about how mail in ballots were supposed to be marked, counted, and when they were valid. The PA courts changed those rules, and they ran the election on the changed rules. So the election in PA was not run in accordance with the State Legislature's wishes. The PA Supreme Court
could have said "This law is unconstitutional [their enumerated power], Legislature come up with constitutional laws before the election." but TX is asserting that the PA Supreme Court (and other non-legislative people and bodies in the other states) don't have the authority under the COTUS to just make new elections rules.
This seems to be a new argument, distinct from the previous cases I've read about that have been thrown out. Since we are a union of States, another State would seem to have a different standing as far as being wronged by an extra-constitutional election. TX actually cites
Bush v. Gore as pointing out that a departure from the legislatively established election rules would present a federal constitutional question.
As far as belonging in state court Article III, Section 2 of the US Constitution states:
In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction.
(bolding mine)
