Underlying this question is an assumption that the writers of the Federalist Papers had an accurate understanding of history. Hamilton, Jay, and Madison were obviously highly educated men, but something that becomes clear when studying the Revolutionary period is the extent to which its intellectual leaders had incomplete understandings of history, shaped by conspiratorial thinking that saw usurpation by elites and political decline around every corner.

Conspiratorial thinking flowed as a deep undercurrent in the river of Revolutionary era thought. To many of the era's writers, Britain was under assault from a cabal bent on destroying the British constitution and enriching themselves at the expense of the people. If it's not clear what or who I mean by this ministerial cabal, well, that's because Revolutionary era writers weren't all that clear about it. There was simply a vague sense that someone, somewhere was trying to manipulate Parliament and British political institutions to their own greedy ends. One popular target of colonial ire was Robert Walpole, considered the first Prime Minister of Great Britain. Other targets were members of the aristocracy who were believed to be bending Parliament to their will. Whether or not these accusations were accurate is besides the point; the point is that Revolutionary era writers strongly believed them.

All of these conspiracies, naturally, drove the intellectual engine of Revolution; it was feared that Britain was already irrevocably on the path towards complete destruction of individual liberty, and that America would be close behind. And this brings us to the Founding generation's understanding of history. The political writers of the time found historical parallels to their situation--of shady cabals bent on destroying freedom for personal gain--in Poland, Denmark, France, and above all, ancient Rome. Their understanding of these events was often incomplete; it is hard to credit their simplistic belief that the Polish or the Danish were a "free" people until all that was suddenly taken away by a self-serving aristocracy. There is also a remarkable Orientalist view that India, China, and the Ottomans had always been under such despotic rulership and had never experienced freedom. Nevertheless, these writers held these (distorted) historical parallels deeply in their hearts, and these served as a kind of argumentative vocabulary as the writers thought about their situation and eventually made the case for independence.

For these era's thinkers, the mere existence of the ephors and gerousia, which were in some sense popularly elected, was enough to make Sparta a republic. It is important to remember here that the "free" Britain these writers lamented did not have universal suffrage in any sense of the word, and neither did the America they envisioned (universal white male suffrage would take until the Jacksonian Era, and in some states even longer). So the highly limited nature of "suffrage" in Ancient Sparta would not have looked so strange or undemocratic to them. Finally, it was enough for Britain to be in the grips of a sinister ministerial cabal for it to be descending into despotism. Never mind that Parliament still existed, and was in some sense popularly elected; it had, to the minds of Revolutionary era thinkers, become fatally infected with despotism.

Sources:

The Federalist Papers

Bernard Bailyn, The Ideological Origins of the American Revolution