I think the practical effect, if not intended design purpose, of a republic is to launder responsibility. To create a layer of indirection and uncertainty between the people who have power and the popular perception of who to hold accountable. The wealthy write the rules and use politicians as patsies. In return for their service to the elite, politicians are offered some privileges and a degree of protection from the angry mobs. The mobs are made to believe that the most effective way to effect change is to vote in new politicians, allowing the old ones to peacefully retire. The economic elite rest easy, knowing the new politicians will serve their interests just as the old ones did.

Sometimes a renegade politician who earnestly has the interests of the common people gets voted into power, but the 'damage' such a renegade can do is regulated by term limits (and sometimes assassination: https://en.wikipedia.org/wiki/Gracchi) Other times, the public believe they are voting for such a renegade, but accidentally empower a tyrant who aims to usurp the elite and have true power for himself. But by in large, a republic regulates the system, maintaining the status quo to the benefit of those who already have power in the status quo.