Why should people support genuine democracy in certain Muslim countries like Saudi Arabia and Egypt?

Majority rules style Athenian democracy was a big time failure. Roman style republicanism was a big time failure, too. So our founders created a system of checks and balances where democratic and republican elements can use majority will while still protecting rights of minorities against majorities and rights of the individuals against the state. When we speak of exporting "democracy", we're really speaking of exporting systems of checks and balances.