Firstly, the Federalists were not the GOP's precursor, given that the Federalist Party died off in the early 1800s

It was the Democratic-Republican Party that eventually fractured into the Democratic and Republican Parties.

Secondly, during the French Revolution, the Federalist Party was in support of Britain against France, whereas the Dem-Republican Party was in opposition to Britain against France, hence why Jefferson even named it the Republican Party in the first place: he felt the Federalists were too supportive of aristocratic policies, and he formed the Republican Party (later Democratic-Republican Party) as a response against them.

Even the GPT users acknowledge this...