This principle does not specify anything about the mechanism that determines how certain executive or legislative positions are filled.
Any such position could be filled for example based on the result of a majority voting by all citizens, or by representatives of all citizens, or based on a random choice between all citizens.
These and also many other possible mechanisms are all equally democratic, because any citizen can fill those positions and all citizens have equal influence on who will fill the position.
On what basis do you claim it means that?
not only objectively impossible to measure but there is no evidence for this. It's Churchillian cliche which has now become a thought-destroying mantra.
I'd almost certainly prefer a mechanically determined pool (say, all Walmart store general managers as of 2020 who are natural-born citizens, >35, not felons, etc), and then randomly pick the President from that pool. Maybe let people use negative voting to eliminate 10-50% of the pool first.
Unfortunately, among recent US presidents, there have also been many who have been much worse than it can be expected for a randomly chosen American to be.
I do not believe that it is possible to make with any certainty an estimate of which president selection system would have been superior, i.e. how the better and worse presidents would have been compensated in the 2 systems.
So judging from the US historical experience, it is impossible to reject the system of a random selection of the president, like for jury members, as clearly inferior to a president voted by the people.
Maybe a combined system would be better, with a set of presidential candidates chosen randomly from all citizens, one of which is elected by popular vote.
This would exclude the people addicted to power who have progressed inside their party through dubious means, becoming the party candidates.
(One of) The problems with political representatives is that they _want_ to be politicians. Either as a means to an end, or as a mechanism to better serve themselves or others like them. On more rare occasions you find the occasional altruist.
Since there's a selection bias going on, how can the current crop act as representatives of their populace when they aren't _like_ those they claim to represent?
It is an idea that is usually so quickly dismissed with "what if we pick a total idiot!?" - as if people optimizing for campaigning as opposed to governing couldn't end up being total idiots as well.
Historically, people have fought to select leaders. Such a system puts the strongest and most brutal in power.
Now we let people talk and then vote. It's a mechanism for resolving conflicts that has been extremely successful at its job.
This is, btw, why freedom of speech is so important and why it's exactly whose we least want to hear, who must to be able to speak. We resolve conflicts with people we have conflicts with, not people we agree with. Talk about it or fight about it. Talking has led to, probably, the most peaceful period of human history.
Why do we have one absolute ruler? You could make up reasons about deciding votes and stuff, but clearly it's a throwback to the good old days when one person killed anough people to become King.
They could work shifts and always have someone awake, while still being able to go on holiday.
Similar applies to CEOs, why does it have to be one person representing the will of many.
There are a lot of modern countries who claim to be republics and even USA has a "Republican" party, but all these modern uses of the word "republic" are wrong, because none of them uses this principle of filling all executive positions with multiple people, so that none of them will have all the power of those positions.
The modern states which use the word "republic" justify their claim by having elected leaders, but that was not something specific to the Roman Republic. There were plenty of ancient states with elected leaders and with various other lesser public functions filled by elections.
What distinguished the Roman Republic from all others was the requirement of electing multiple people for all positions, including for the supreme executive function, for which they had 2 consuls, and not 1 president.
There's many logical reasons. The federal bureaucracy handles the day to day work of federal agencies, not the actual individual elected to be the President. The President appoints a cabinet to oversee those agencies. So there's no need for any individual to be awake 24/7 making decisions which means there's no need for multiple shifts of executives to be available.
Multiple executives would actually be a terrible system as situations last longer than a single shift. Day Shift President would have to hand over a bunch of context to Swing Shift President for any issue lasting longer than a single shift. Swing Shift President may not agree with Graveyard Shift President on the best course of action which leads to an inefficient and/or legally arduous process of adjudicating primacy of authority.