You had a typo (you meant that the VP is * not* a Senator.
The pre-12th-Amendment system for electing the president and VP sucked, with one of many problems being that if two candidates tied for first place in the Electoral College (with each receiving votes from a majority of electors—remember, each elector got two votes), the House would elect the president and the loser of the vote would become the VP, without the Senate getting a say. Had the lame-duck Federalist House in 1801 refused to elect either Jefferson or Burr so that the Federalist President pro tempore of the Senate got to act as president of the U.S., it would have resulted in a lawsuit claiming that the Succession Act of 1791 was unconstitutional because it placed persons other than Executive Branch officers in the line of succession (an argument that James Madison had made on the floor of the House back in 1791, but the Federalist majority had ignored him). The House rules also provided that no business could be carried out until the House had counted the electoral votes that elected a president or, if the Electoral College failed to elect, until the House elected a president. There really wasn’t much to be gained by Federalists to refuse to exercise their duty to elect a president.
The 12th Amendment does not say whether, in a contingent presidential election, the vote of a majority of the members of a state’s House delegation for a particular candidate is required, or if a plurality vote of the members of the delegation would be permitted. When the Constitution is silent (in text or structure), the rules of each house may fill in the gaps. I believe that in 1824 the House rules required a majority vote by a state delegation in order for the delegation’s vote to go to a candidate, but, if there was a contigent election today with three or four candidates and there a deadlock after several ballots, I think that we might see the House change its rules to permit a plurality vote from a state’s delegation to be sufficient; however, one thing that the House could not change is the requirement that it would take the vote of 26 state delegations to elect a president.
As for the provision of the first Succession Act calling for a new election, I think that it was unconstitutional. Article Ii of the Constitution provides that the presidential term is 4 years, that the VP replaces the president in case of his death, removal or incapacity, and that Congress shall determine by law which officer shall act a president in cases of vacancies in both offices or of incapacity of the sole president or VP in office. Article II does not empower Congress to call a new election, only to legislate for the officer to act a president in the cases described above, and the fact that Article II begins by saying that presidents are elected to four-year terms is a clear indication that tge Framers did not contemplate special elections for short periods.
I'd have been a Federalist but that was very shortsighted of them. All because they didn't want Jefferson to be 2nd in line.
It was equally as shortsighted of the 80th Congress to put the Speaker and President Pro Tem BACK in (after they were removed by the 1886 act) during the early Truman administration with the Vice Presidency vacant.