In the true sense that the people are the state, a democracy would hold that the people can push through changes to the system that might not be in the interest of those who currently hold power under the very system the people are trying to change.

And yet... how does a population convince itself that elections are conducted fairly?

Any change proposed in good faith is distorted by anyone who needs things to stay the way they are. Fear, uncertainty, doubt.

No age limits, anyone (including non-citizens) can vote, and there's no registration system?

And people can vote as many times as they want, in as many districts as the like?

All of these are current state controlled processes which restrict the population from voting.