From my understanding, according to the 12th Amendment, if there's a tie in the electoral college, then Congress chooses the president/vice-president, with the House selecting from the top 3 candidates who received at least one electoral vote, while the Senate chooses the veep from the top 2 candidates. If this is correct, then what was the logic behind the different numbers? Wouldn't it make sense to have both chambers vote for the top 3 candidates for the respective positions?
The 12th amendment was meant to solve several problems with executive office elections. One, it was meant to avoid deadlocks as much as possible. Two, it was meant to avoid the problem of having the president and vice president be staunch political opponents. If there's a majority of electoral votes for any one ticket, then everything goes well, and the second problem is also avoided because of the switch to an executive ticket versus 1st/2nd place for potus/veep.
But if there isn't a clear majority then there's the potential for everything to go sideways. So the 12th amendment is also meant to avoid the calamitous possibility of failing to have a president when the previous head of state vacates the office. That's why the veep is chosen from only the top two candidates, because that results in no possibility for a tie. Remember, of course, that the Senate will always have an even number of members, but the president of the senate (the currently sitting VP typically) can cast tie breaking votes.
That means it's not possible to have a question of who would be the next vice president, it's one and done, two choices, no possibility for a tie. And in case congress has failed to choose a president by the time the next term begins, the VP becomes the president. Which encourages congress to pick a winner but also provides a backup plan in case voting is so contentious it drags on for months.