So I think he took a chance to die for the Union by removing the greatest threat of secession once and for all.
Have you ever considered the possibility that it was Hamilton who was actually the greater threat to the future of the United States? Of all the major founders, none was more reviled by his peers than Hamilton. And that includes the next four presidents. Not just Jefferson, but Madison, Monroe, and even John Adams all had one thing in common - none of them could stand Hamilton. All generally concurred that he was hot-headed, had questionable morals, was exceedingly ambitious in his political machinations, and possessed a highly manipulative character.
So perhaps it wasn't Hamilton that saved the union by tarnishing Burr, but Burr that saved it by eliminating Hamilton.
“Manipulation”, “hot-headed”, “questionable morals”, “ambitious”? At least one and often more of those labels fit the presidents you list.
Very few of the Founders could stand against Hamilton in debate. None worked as hard or rapidly as he. Jefferson called him a “colossus”. There were more who admired him beyond all but Washington than hated him. Virtually everyman who became his friend stayed his friend for life.
NOTHING mattered more to Hamilton than protecting and strengthening the Union and none other than Washington did more to create it.