Jefferson's election was often referred to as the "Revolution of 1800." Jackson's election in 1828 was also seen as a radical turn of events. As you say, there was much difference between the ideas that Hamilton or Adams had about the federal government and those that Jefferson represented. The old Federalists who supported the Washington administration were increasingly alienated in Jefferson's and then in Jackson's America. Some second generation Jeffersonians were similarly estranged by Jackson's victory, though their more radical fellows might have embraced him.
The Civil War did mark a break, but was it so catastrophic a break as to constitute the creation of a new Republic? In some of its policies, Lincoln's America represented a return to Washington's and Madison's view of government after the long Jacksonian hiatus. In other ways it was truly radical, but the radicalism wasn't necessarily something the Republicans intended to impose on the country in 1860, but was an outgrowth of war, emancipation, and the defeat of the South.
One major reason why Lincoln's America looks so different was the rise of industry. Different sorts of people came to power after 1860. Politics in an industrial nation will always look and be different from those in a more rural republic, but it would be a mistake to conclude from that that Jefferson's or Jackson's views on the Constitution were the same as those of Washington or Adams or that it was Lincoln's ideas about government that destroyed their America.
If the "Second Republic" did perish it's likely that it did so in 1860 with secession. The idea that those who destroyed the old union represented the "real Constitution" is an illusion. They felt that the Constitution of 1789 didn't provide them with sufficient protection for their interests and that a new departure was necessary. Whatever came after 1860 was bound to be different, regardless of who prevailed in Congress or on the battlefield.
It's also worth noting that when Republicans Jefferson, Madison, and Monroe were in the White House, they adopted some of the Federalist policies that they had criticized when they were in opposition. Lincoln could cite some Jeffersonian Republican precedents for his policies, and he and his idol Clay never particularly thought of themselves as Hamiltonians. Jefferson himself was a swing figure between constructive statesmen like Madison and Monroe and more radical figures like John Randolph and John Taylor.
If Jefferson's reputation has fallen in recent years one reason surely is that he was all over the map: pro-slavery but anti-slavery, libertarian but willing to countenance nationalist and restrictive policies, a revolutionary and radical with a conservative side, an agrarian isolationist republican who put the country on the road to empire, an egalitarian aristocrat. The many-sidedness that made Jefferson such an attractive, provocative, and intellectually fruitful figure in previous generations, now looks like something of a muddle and a mess. There's still a wealth of ideas and perceptions in Jefferson, but one can't blame people for throwing up their hands.
Lincoln could cite some Jeffersonian Republican precedents for his policies, and he and his idol Clay never particularly thought of themselves as Hamiltonians.
Clay was elected to Congress the same year that Hamilton died, and he immediately picked up the fallen standard of Hamilton with respect to the tariff and Hamilton's vision of a nation built on manufactures. Clay's "American Plan" was but an extention of Hamiltonian principles.