How do the Federalists relinquish power in 1800 if the country is not "democratic" by then?
The Federalists lose because South+Middle outweighs North. I don't see what that has to do with whether the government is popularly elected or not.
Even if you argue that the South and New England were undemocratic (going along with the interests of the planter and merchant class, respectively), which is of course ridiculous, clearly the Middle voted Republican only because it was democratic. New York was Republican precisely because Burr built a democratic coalition there.
If we are talking about presidential elections, then New York was one of the last states to move to the popular vote. On the other hand, the legislature that selected the presidential electors was democratically elected, so the franchise is probably the more relevant factor in determining whether an election is democratic. But even there, New York didn't completely abolish property/tax qualifications for the white vote until 1821.
The two don't always go together. RI was one of the first to move to the popular vote in the selection of presidential electors - as early as 1800 - but still retained some property qualifications in the early 20th century when the South was reinstituting them.