Who says the world would be a better place if there were no poverty, maybe the rate of crimes might reduce but there would still be crimes due to greed and hunger for more, the wants of humans is insatiable so there would still be competition and quest for superiority which would lead to crimes, remember that not everyone who commit crimes are poor people so eradicating poverty won't stop crime totally.
I even think there would be chaos, imagine a world without no government, no hierarchy, no superiority everyone are the masters of themselves it would be very weird, I think the rich and poor class exist for the world to be balanced and if everyone was equally rich there would be chaos.
The rate of crime might not even reduce because you know people are not always satisfied and contented with what they have, you will see some people stil conniving with people of like mind to extort or still from other especially if they notice that some persons don't really the mind to do anything, you even reminded me about the world without poverty and everyone is been allowed to do what they like, Sodom and gomora would've been better than the world of today, even with security agencies everywhere we still see things that sometimes go well in humans eyes happening in this world, I concur to all you said.