In a few places Americans have run off elections if no candidate gets a MAJORITY of the whole vote
The founders clearly understood the absolute majority concept.
How Different would the US be.
Obviously smaller parties would not be as disadvantaged as in OTL.
How different woudl 1824 have been ( I am assuming the electoral college exists but that there could be second ballots in states where no candidate won half the votes cast.
In those states which chose electors by popular vote (there were still six in which the legislature chose them) there were only a few genuine three-cornered contests, so that the winner usually had an absolute majority. The exceptions were -
Maryland - Adams 14,632 (3EV), Jackson 14,523 (7), Crawford 3646 (1) Clay 695.
Ohio - Jackson 18,457, Adams 12,280, Clay 19,255 (16)
Indiana - Jackson 7,343 (5), Adams 3,095, Clay 5,315
Illinois - Jackson 1901 (2), Adams 1,542 (1), Crawford 219, Clay 1047.
Ohio doesn't change, as most of Adams' votes are likely to switch to Clay.
Maryland - Crawford and Clay votes probably switch to Adams, so he gets a larger share of the electoral vote. Hard to say how much.
Illinois - Probably no change. If
all the Clay and Crawford votes switch to Adams, it might be 2-1 the other way, but it's a long shot.
Indiana is the one that makes the difference. Adams' votes will likely switch to Clay, and it's winner take all, so he picks up five EVs from Jackson. Combined with his losses in Maryland and maybe Illinois , this drops Jackson into the low 90s, while Adams' gains bring him into the upper 80s, but neither is anywhere near a majority so this doesn't matter. What
does matter is that Crawford is unchanged at 41, while Clay has moved up to 42, and just pipped him for third place. So the HoR will be choosing between Jackson Adams and Clay. Given Clay's influence in the HoR (he is its Speaker) he is well placed to win. Certainly if Adams can't, his states will prefer Clay to Jackson. Sounds like an interesting WI.
I understand that Lincoln won more than half the votes in enough states to win in 1860 whilst of course polling almost no votes in the south
That is correct. Run-offs would probably have given California (4EVs) to Douglas and Oregon (3) to Breckinridge, but Lincoln still has a comfortable majority.