I don't really have a serious answer for this one, because the 12th
amendment would have to be passed as partisan politics came along in
the early 19th century, if not as early as 1800. Had the amendment not
been passed prior to 1824, Congress would definitely have done that
after that troubling election.
History would have been very different from that point on had the 12nd
amendment never been enacted. So there's no point to think about what
the presidential election would have looked like in 1960. The
manifesto destiny might not have gotten as far as California because
of the lack of the 12nd amendment, and thus there would be no Dick Nixon!
My most informed guess is that the Union would not have survived the
Civil War. There would simply be a lot of presidential (and
vice-presidential) assassinations right before and during the War and
the control of the White House would be switching between the
Republicans and the Democrats. (Note that Andrew Johnson was chosen
for Lincoln's second term, when the Civil War was by and large over.)