Captain Shatner wrote:
Obviously, the Federation as a whole is a democracy founded on American ideals, as demonstrated in the episode "The Omega Glory," Kirk can even recite parts of the Constitution!
Again, not really. Due to widespread pop culture, I can quote parts of the US Constitution, and I'm not even American nor I live in a country founded on "American ideals".
It would be useful if you defined what you mean by "pure democracy".
When I first read the original post, I almost asked the same question, but I looked up the term first:
So, in a pure democracy, every citizen has an equal say in every vote. Extremely impractical, as you can guess. I don't think it's ever existed in the real world. In practice, we have varying flavors of indirect democracy.