A serious blemish in an otherwise decent piece.
Jefferson's Democratic-Republicans were the closest thing the country has really had to a libertarian Party (but not Libertarian Party.) They surely aren't the forerunners of the modern Democrat Party. Indeed, I would even argue that Jackson's populist version of the Democratic Party -- which is closer -- still had little in common with modern Democrats.
The statement is wrong [and worse from an editor's point of view, it is gratuitously wrong, since it has no real bearing on the piece. If the author is trying to evoke sympathy from present Democrats by pointing out their kinship to Jefferson, he is wasting his time.]
That I would disagree with. Jackson was the first to court the low information voter, and the Democrat Party has thrived on that to their own advantage for the last 170 years.