This is a straw man. Nobody, including the Founding Fathers, claimed the original Republic was ideal. The Constitution's vagueness and incompleteness were widely admitted at its signing.
Pure democracies don't work. They devolve into populist, majoritarian nightmares that tear themselves apart. This has been a consistent failure mode across over three millennia. We have minority-protecting, stabilizing features built into our system that have been systematically dismantled and attacked over the past decades, all in the name of more democracy. That more democracy isn't always a good thing has been absent from the discussion, in part due to Cold War and ensuing dynamics, in part due to ad hominem rejections like the one you made.
Suppose that the US were to create a constitution today, based on today's needs and the anticipated needs of the next 50 years. How different do you think it would look from what was written in the 18th century? (Put aside the impossibility of doing this in today's political climate, as that's not the point of the question.)
It starts to sound demi-religious. You shall not disobey the word of a larger power/figure.
And then society starts to rift over time, through the interpretation by different sects about what the words meant.
Over centuries you get a pretty twisted view of the original values, interpreted by a few different sects, and an ever growing wedge between those interpretations that gets harder and harder to reconcile.
Just like most major religions.
It's called Empire of Liberty. It's about the first 3 decades of the US and definitely shows how much was not planned at the time and how much of the constitution was intended as just a starting point to be edited soon after. Very much not the perfect document people treat it (knowingly or unknowingly) today.
In contrast, states in which the rulers are elected from among a limited set of candidates chosen by the government, do not qualify. For instance, some states curate the selection of candidates by jailing or murdering opposition candidates. Others vet their candidates through a separate body of religious leaders.
In the US, gerrymandering probably does not pass a basic constitutional sniff test, but getting governments to relinquish powers that they enjoy is no small feat.
Anyway, my point was that "republic" effectively means "liberal democracy" in this specific context, where someone invokes the Founding Fathers and contrasts it with direct democracy. And when someone talks about present-day democracy, they usually refer to the entire Western system rather than to simple majority rule.
This has been known since antiquity, and is why the Founding Fathers designed not a democracy but a republic that combines elements of monarchy, oligarchy and democracy. (The three branches of government.) That we’ve forgotten this lesson is a likely contributor to the current instability.