I discussed this in soc.history.what-if back in 2000. I'll repost it here, though I am more inclined now than then to believe that there would have been a Second Missouri Compromise without Clay:
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In 1809, Henry Clay was wounded in a duel with Humphrey Marshall,
leader of the Federalist party in Kentucky, who had called Clay
a "poltroon." (Marshall btw was the brother-in-law as well as a cousin
of John Marshall.) Suppose Clay had been killed.
A few possibilities:
(1) People sometimes forget that there were *two* Missouri Compromises--
one of 1820 and one of 1821. Clay had an important role in the first
(although it was Senator Jesse B. Thomas of Illinois who first
suggested the 36 degrees, 30 minutes line) but the second (which was
prompted by Northern outrage at the new Missouri Constitution's
prohibition of "free negroes and mulattoes" from entering the state)
was almost entirely his work. (The second compromise, which basically
said that Missouri had to promise not to abridge the privileges and
immunities of any citizen of any state, only got through the House by a
vote of 87 to 81.) One can always say "Someone else would have worked
out a compromise, the Union would not have fallen apart"--but I am not
so sure. It is hard to think of anyone else who would have had the
power over the House Clay did.
(2) Of course other people would advocate tariffs, internal
improvements, etc. But would others be as successful in tying them
into one program and giving it a catchy name like "the American System"?
(3) Without Clay, if the 1824 election went into the House (assuming
the other candidates to have been the same as in OTL) it is hard not to
see the Western states going for Jackson, which would probably lead to
Jackson's election, unless the Crawfordites were more stubborn than I
think they would be.
(4) Assuming there would eventually be some kind of Nullification Crisis
(there would have been protectionists who would have gotten a high
tariff enacted even without Clay, and resentment in much of the South,
especially South Carolina, of that tariff) I am not sure that anyone
other than Clay would have had the authority (because of his
association with the American System) to get Northern protectionists to
agree to the Compromise Tariff that helped to end the crisis.
(5) Clay may actually have been an obstacle to the unification and
strengthening of anti-Jackson forces, because (a) as a Freemason he had
a hard time with the Antimasonic vote, and (b) he was haunted by
the "corrupt bargain" charge. Eventually, an anti-Jackson party would
have arisen, whether it would have called itself "Whig" or not (Clay
didn't originate the use of "Whig" for the anti-Jackson party, but he
did much to popularize it). But obviously its history would have been
very different, especially if Jackson had first been elected in 1824
rather than 1828. OTOH, if Clay had not been its early leader, who
would have been? Webster? He was too aristocratic, too Federalist to
be very popular.
As for still later events, like the Mexican War and the Compromise of
1850, things might have changed so much before then that I don't even
want to consider them yet, though others are welcome to do so.
https://groups.google.com/d/msg/soc.history.what-if/04pe5Lqf-jY/1Uwh9ky-0pUJ