The question presents a false dilemma. Religion arose in early humans and primitive societies as an expression of the need to make sense of the world, explain the unexplainable, provide assurance about the future, and to provide a common basis for cooperation to make life better. Religion is a part of both human and societal evolution. The process of natural selection does not reward what is good or bad, but what works.
Thankfully, human evolution includes the capability of reciprocity, empathy, and altruism to balance self-interest, anger, and vengeance. Early in human history it was found that when people restrained their violent and selfish interests so others would not retaliate they could cooperate to make societies that were safer and made life easier for all involved. The better the cooperation, the more that could be accomplished. These principles were expressed in theistic religions, non-theistic religions, philosophies, and governments. In many societies, religion and government were indistinguishable from one another.
So the question really is, “Can humans create a better world without a basis for cooperation?" No.
Does it matter what that basis is? Only insofar as the vast majority being willing to accept it.
Is religion the best basis for cooperation? Well, that is the crux of the matter.
Given the many varying religions, theistic and non-theistic, and the growing number of people who have no religion and who lack any belief in God or the supernatural, I would say the best basis for cooperation is human values upon which everyone can agree. Archeology and history both indicate that the common ethical elements of all religions and governments are reciprocity (treating others as you would be treated), empathy, and altruism. Reciprocity motivates people to treat others decently, and governments make laws to prevent its citizens from being harmed by those who violate that principle. Reciprocity also promotes cooperation: if we all agree to drive on the right side of the road it helps everyone travel more efficiently and safely. Empathy involves understanding the situation and needs of others. Altruism motivates people to help those in need and to help improve living conditions within a society.
So, I would say that the best basis of cooperation on which the world can become a better place would be founded upon reciprocity, empathy, and altruism, regardless of anyone's religious beliefs or lack thereof.