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.
You are right, the election of 1796 did result in Adams/Jefferson. But even though Adams and Jefferson were political enemies (they later reconciled and oddly died on the same day in 1826-50 years after the Declaration of Independence) the 12th amendment was proposed after the disaster of 1800. Adams and Jefferson probably spoke little but I don’t think that anyone thought that Jefferson would try to do Adams in. But Burr showed the fatal flaw in the original plan. Washington thought we’d all be Americans (what a quaint thought—remember right after 9/11 when we kind of thought that again?) but the dark side of human nature reared its head. Since the Electoral College was designed with Washington’s ideals in mind, it didn’t take into account the passions of political parties.