the same people doing menial tasks would also be rotated into administrative tasks
Isn't this making the assumption that someone who can perform the former, can also perform the latter? It very obviously would prove chaotic for the business as multiple individuals, with their various management styles and skill sets, rotated in and out of the leadership position.
Giving "everyone a turn" sound like sometime out of a preschool daycare.
Different societies/cultures/economic systems will use different methods. Does that make your's or their's superior?
The sign of a superior system, is one that has been shown to work over a extended period of time. Our current system might not be perfect, and it might not make you feel warm and fuzzy inside, but it works. Even when it occasional get knocked back a few steps, it recovers after a few years and continues onward.