In the historical context I would see "Vox Populi, Vox Dei" as intentionally juxtaposed against the "Divine Right of Kings" (the leading political philosophy in Europe for the last few thousand years).

I think there is a TON of interesting discussion to be had on the best ways to _implement_ democracy, but locating the source of political power directly in the people who are governed seems like a pretty solid idea to me... I have not been able to come up with anything better.

Indeed. Democracy is "least bad" because it at least allows people to choose, but don't be surprised if they make the wrong choice. Evidently one problem outweighs the other, and for advocates for democracy it's that the latter issue is outweighed (the former is that the people have no choice, this is people making the wrong choice). If men were angels, a nominally totalitarian government could be absolutely ideal.