There's one last consideration; Art. 1, Sec. 5 leaves the final judging of a Senate election up to the Senate itself. There's no similar clause for the appointment (currently through popular election) of the Electoral College.
Oh yes, there is:
Amendment XII-"they shall make distinct lists of all persons voted for as President, and of all persons voted for as Vice-President, and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate; The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted..."
In the event that the Joint Session of Congress cannot agree on the counting of the votes (for example, if multiple sets of electors are presented from the same state), Congress can follow the Electoral Vote Counting Act, or it can make new law to deal with the situation.
In either event, Congress in Joint Session with the President of the Senate in the chair is the judge of electoral votes-NOT the USSC.