A state or states can simply not certify, so where does Amendment #12 go from there?
Correct, but the provisions referred to in the 12th amendment don't really come into play unless no one wins a majority (270) electoral votes in the electoral college.
If after the states have held their elections to select electors, and the electoral college electors cast their votes at noon in their respective state capitols, and there is no candidate with 270+ electoral votes, then the election gets thrown to the incoming House. The House then votes between the candidates with the 3 highest electoral vote totals. The easiest way to get a result where no candidate wins a majority of electors is if there are at least 3 candidates who win the popular votes in states and the candidate with the most votes does not gain at least 270 electoral votes. Another if less likely way is if there were a 269-269 tie in the electoral college.
If it were already known before the electors meet in their state capitols in December that no candidate would get an electoral majority, there might be less discipline amongst electors in voting for the candidates to which they are pledged. In the case of a 269-269 tie based on the popular vote on election day, just 1 elector could cause the election in the House to be a 3 way race by voting for another candidate. What if an unfaithful elector from Vermont decided to vote for Bernie Sanders instead of Hillary Clinton? The projected vote would then be Hillary Clinton 268, Donald Trump 269, and Bernie Sanders 1. The race would still go to the House, but Bernie Sanders would be one of the 3 top candidates and he could receive votes in the House. Since the electors meet at noon on the same day in their respective state capitols, any unfaithful elector in the eastern time zone could have a ripple effect in other states that are in the Central, Mountain, and Pacific time zones. Knowing that the election will be sent to the House anyway, some electors in states in time zones to the west could vote for other candidates to knock Bernie Sanders out of 3rd place.
Votes aren’t counted until January.
On the other hand, you’re right - sort of.
In 1960, an elector pledged to vote for Nixon offered to vote for Harry Byrd if Democrat electors would join him.
So far as I know, it’s just honor system - the faithless electors would not know for certain that the bargain was kept.
The rules were different then but the EV in 1800 was:
Jefferson 73, Burr 73,
Adams 65, Charles Pinckney 64, Jay 1.
Had one of Burr’s electors voted for someone else, Jefferson would have won the Presidency outright.
Actually that did happen. See footnote d.
https://en.wikipedia.org/wiki/United_States_presidential_election,_1800
The second vote should have been spoiled, and Burr should have been elected President (outright) with 73 votes, and Jefferson Vice President (outright) with 72 votes.
Was the argument made, I wonder?