I get your point, but I would contend that the present-day GOP is every bit as much a descendant of the Democrat-Republicans as the present-day Democrats are.
Possibly a nit, but I think it’s a significant one. The Democrats, Whigs and Republicans are (were) all branches on the same American tree.
Actually, that’s not true anymore, although it was up till the late 1800s, when the Progressive movement made major inroads into both parties. The Progressives get a bad rap among today’s conservatives, much though not all of it deserved. But it is pretty difficult to see them as anything but the introduction of foreign, French Revolution based ideologies into American politics.
Both parties are presently uneasy coalitions of groups attempting to adhere to the principles of both Revolutions. With liberal Democrats mostly French in their ideology, with their left wing adding a dose of Russian and/or Chinese revolutionary fervor. Meanwhile conservative Republicans are mostly though not exclusively devoted to American Revolution ideals. And the squishy middle an unstable mixture of both, though few of them are aware of it.
The problem is that the French (not the mention the Russian and Chinese) and American Revolutions are inherently incompatible.
The French Revolution taught that The People were sovereign, with individuals having no rights at all against the General Will of The People, which of course was by definition subject to interpretation and application by whichever group of thugs was temporarily in power.
The American Founders believed that the people (lower case) were sovereign, but that as an entity or group they had no right to trample on the inalienable rights of each individual making up the people.
When the old Federalist Party collapsed after the 1816 election, Northerners joined Democratic-Republicans for the 1824 election, and just managed to elect John Quincy Adams.
But unity didn't last long.
By 1828, Adams ran (and lost) as a National Republican, a party which also fielded candidates in 1832.
By 1836 National Republicans had become Whigs, a party which lasted through 1852.
After Whigs collapsed, abolitionist Republicans became the main anti-Democrat party.
So the sequence was:
So, from the Founding of the Republic until Goldwater in 1964, the "Solid South" was solidly Jeffersonian / Jacksonian Democrats.
During that same period, the North was represented by Federalists, National Republicans, Whigs and then Republicans before turning, in the 1960s, solidly Democrat.