Because I would have to disagree if that's the case a system of government makes a big difference. Look st the divide in America for starters.
I believe America's issues go far, far beyond their system of governance. And the division you speak of isn't exclusive to the USA. Even in Australia, there is a substantial right-wing vs left-wing division; and yet we're a constitutional monarchy?
Further, it's important to remember that there are a number of different types of republics Australia could become. Obviously the USA system involves a directly-elected president, however Australia does not have to follow suit. Personally, I believe the model recently proposed by Jones and Pickering (
https://theconversation.com/a-model-for-an-australian-republic-that-can-unite-republicans-and-win-a-referendum-89919) would be very effective in Australia; as it would retain the role of the Executive as being generally ceremonial (thus maintaining stability), whilst also ensuring that this person is:
1) an Australian,
2) someone who has outstandingly served Australia in some way (like an Australian of the Year), and
3) chosen by the people through a non-binding vote.