There is one outright falsehood on this video: Rome never became a democracy. It went from republic to monarchy.
I think the video is generally wrong regarding democracy. As the example of Solon shows, a democracy with a fixed body of laws, essentially a constitution, would be just as much ‘rule of law’ as a constitutional republic. One of the key reasons this didn’t work in ancient Greece was that elected officials didn’t abide by the constitution. However, this isn’t a failing of democracy, just as American politicians who don’t abide by our constitution don’t represent a failing of our republican form of government. The same problem is inherent in both systems: They depend on those in power respecting the law.
However, democracy is not really a viable option for a large state. It depends on every eligible voter being able to gather to discuss and then vote on proposed laws. The Greek city states could do this because each city was effectively its own nation, and because they limited the vote to landowners, etc. Historically, the US never could have achieved a democracy. Even in 1789 you couldn’t have gotten all the eligible voters together to discuss. There were just too many of them. (It’s true that it could be done now, via internet, but no one I know if is seriously suggesting that.) No one today is suggesting the US adopt a democratic form of government. When most people say “democracy,” they mean a representative, constitutional form of government.
I do agree that the left-right political spectrum is misleading. The real spectrum is statism-individualism, with totalitarianism on the far left and anarchy on the far right. However, the spectrum should run: totalitarian state, monarchy, oligarchy, republic, democracy, anarchy.