The interesting thing about the 12th amendment is the history behind it. Originally, the Constitution was written with the thought that we’d have no parties like bad old England. So the electors were to vote for two men to become President. The one who got the most votes would be President and the one who came in 2nd would be Vice President.
But parties did arise. In the election of 1800 Thomas Jefferson and Aaron Burr running as Democratic Republicans won against the Federalists. Jefferson and Burr received the same number of Electoral votes. Seeing his chance, Burr tried to steal the election (sound familiar?) by trying to drum up votes in the House to be elected President. Hamilton led the forces who opposed Burr and later Burr would pay him back by killing him in a duel.
So now the electors vote separately for President and Vice President. Under the old system Hillary! would be Vice President-Elect and wouldn’t that make for an interesting four years.
That happened in the 1796 election. Many of the Electors refused to vote for the running mates, voting for other people instead. The result was John Adams being elected President and Thomas Jefferson, Adams's very hostile political opponent, being elected Vice President.
The 12th Amendment was adopted to prevent ticket-flipping. This became possible after the two-party system was established. Jefferson realized that the Federalist Party hated him so much that it would likely tell its Electors to votes for his running mate in the 1804 election. This would result in his running mate being elected President. His running mate might encourage this, just as his running mate in 1800 (Aaron Burr) had in the House. The 12th Amendment's splitting each Elector's votes between President and Vice President (rather than 2 for President) prevents ticket-flipping.
If the original Electoral College system applied to this election, the Democratic Electors could vote for Mike Pence. This would give Pence the most electoral votes and the votes from a majority of the Electors. Mike Pence would be President. Trump would have the second most electoral votes and so would be Vice President. Would Pence resign so Trump could be President? The point is that every Presidential candidate wouldn't be able to trust his running mate, who would easily be able to double cross his party and the Presidential candidate. The 12th Amendment allows every Presidential candidate to trust his running mate.
“So now the electors vote separately for President and Vice President. Under the old system Hillary! would be Vice President-Elect and wouldn’t that make for an interesting four years.”
She would be kept “on ice” in some underground bunker in Alaska as the full-time “designated survivor”