> The two most powerful countries with non-figurehead executives (France and the United States) are perpetually beset with bickering and internal fighting instead of "checks and balances".

That's not so much a problem of a Presidential system as a problem of the a Presidential system in a country which also has a poorly-representative electoral system in general (particularly for the national legislature).

However, there aren't a lot of systems that have electoral systems that produce effective representation and Presidential systems in the same place.