I actually had written an answer about the effects of the 12th amendment on the politics StackExchange that details how the original elections worked (or failed) under the old system.

The interesting thing to me about this is that after Washington, there had always been running mates, and the problem wasn't that the President and Vice President may be political opponents. The problem was the old system was open to gamesmanship that thwarted the will of the voters.

1796 did not end with an Adams-Jefferson administration because Jefferson came in second. If things went as planned, it would have been Adams-Pinckney instead. But Hamilton preferred Pinckney over Adams and tried to sway the electors for Jefferson-Burr to vote Jefferson-Pinckney instead, which would have led to a Pinckney-Adams administration despite Pinckney campaigning with Adams as his presumptive Vice President. However, his plan didn't work out, and Jefferson ended up getting the second most number of votes. This led to an Adams-Jefferson administration which was not supposed to happen. This was bad, but the shady dealings happened in the dark, and Adams was at least elected President in accordance to the popular vote. The politicians at the time thought that they could just sweep this under the rug as they now had a better understanding of how to manage their electors.

But, that turned out to be false. In 1800, they planned to be smarter with allocating the electors' votes, but the Democratic-Republicans failed and accidentally cast the same number of votes for both Jefferson and Burr. Under the Constitution, a tie is decided in the House, and the makeup of the House meant that Federalists had the advantage. They preferred Burr over Jefferson, so they tried to subvert the election and appoint a Burr-Jefferson administration rather than Jefferson-Burr. Hamilton ultimately convinced the Federalists to relent and give the election to Jefferson. This was now the second time that Hamilton intervened to orchestrate the results of the election, and this time, it was all out in the open on the House floor. Furthermore, in both of these instances, Hamilton's actions screwed over Burr, leading to the infamous Hamilton-Burr duel that left Hamilton dead and Burr disgraced. So not only did the election show that this Constitution was failing in the democratic ideals of the revolution, it also led to the untimely downfall of two of the country's top political leaders.

So yes, the 1796 election exposed a pretty major issue, and the 1800 election showed that that issue could not be ignored. However, if you're suggesting that the 1796 election led to the 12th Amendment because it showed the problems that arise when the President and Vice President are not politically aligned, I'm not so sure. It's possible, but I don't think that was a revelation to them. At the very least, the parties at the time were always trying to fill both offices with specific people, even before the 12th Amendment. The biggest problem they were trying to address was the way that the old system could be gamed by political elites.

Also, sorry for the big wall of text. I just find this to be a very interesting topic.