Stone Cold
Superstar
No it wouldn't. You seem to believe that religion and science function in terms of a dichotomy, you are wrong. The structures that our societies are founded on and our ethical and moral mores do not stem exclusively from religion. You take away religion, the courts still exist, the apparatus of state that were explicitly separated from religion still exist. Religion may have played a fundamental role in the founding of these principles in a societal context but they became self-sustaining centuries ago.
If this was the case then secular communities would be untenable and this is clearly not the case.
Science would not have to do anything, there isn't any person or any society that claims to live their lives in accordance with science atheist or not. It isn't in competition with religion except in the minds of those that question it under false pretenses.
I think you misunderstood the question(that might be my fault). If religion never existed, would any of that exist?
If religion never existed and theory of evolution was true and everyone subscribed to it, would the world be a better place?