So, how do we know that Burr hadn't already pulled the trigger by the time Hamilton's bullet shot past him? ITTL, rather than Burr dying in the duel instead of Hamilton, couldn't it be plausible for BOTH Hamilton and Burr to be killed in their duel with each other? And what would be the impact of that?
The impact of that would be no Hamilton (so... no changes there) plus no Burr (so... still no Western land scheme/conspiracy with Wilkinson).
But couldn't Hamilton have at least influenced elections for years to come? Plus, what about his Christian Constitutional Society?
The problem is that Hamilton was politically dead by 1804. He'd pretty much alienated everyone, and he had quit the Federalist Party in disgust for considering Burr as their candidate. Then he schemed to get Jefferson elected over Burr in 1800, which most Federalist considered to be a really bad move. so he was at odds with everyone, including his former friends.
Killing Burr in a duel would be one more disgrace. The incident haunted Burr for the rest of his life IOTL; that would be no different for Hamilton. His Christian Constitutional Society would be his
only vehicle for attempting to influence politics, and it would be marginal. Very few people would want to be associated with Hamilton by that point.
Maybe he'd be able to influence state politics in New York, but national influence was outside his reach by then.
Long shot option: he keeps his head down for a few years, makes amends with the Federalist Party, overcomes his earlier disgrace, and by 1812 is respected within the Party again. Arguing against the very idea of secession (a Jeffersonian notion in the first place), his influence manages to prevent the Hartford Convention of 1814. The Federalist Party is not disgraced, and when the war ends, they succesfully blame Madison for it. The Federalist Party continues to exist, instead of fading away.
But that assumes he can regain his standing within the Federalist Party, which is very unlikely.