The ancient Greeks were pioneers in the world of democracy and taught the world that representative government could work with rules of engagement applied. The citizens were the only ones who could vote, and they were wealth landowners, but it was a beginning. Sadly, progress in recent decades in severely lacking.
The Greeks had a lot of different kinds of governments, because there were many different city-states in ancient Greece, and they each had their own government. People's ideas about what made a good government changed over time, too. For the most part, Greece began by having monarchies, then oligarchies, then tyrannies and then democracies, but at each period there were plenty of city-states using a different system, and there were many city-states which never did become democracies or tyrannies at all.
The ability of citizens to vote in free and fair elections is crucial in a modern democracy. Indeed, it is difficult to conceive of a modern democracy without the citizenry being able to vote for their government. Though democracy can be messy, it is, as Winston Churchill was said to have said, "The worst type of government aside from all others."
Greece should not have stayed with their old ways, because we know more as a society now. We now know that it's important to make sure that everyone has a voice. Greece was right to update their political system in ways that are more inclusive. It made their society better.