What if no one worried about whether or not there was a God? What if people were just themselves--people, and accepted one another for what and who they are? Would not the world be a better place? Without all the hang-ups, would we not be more human and humane?

Its not really a viable question. People take to religious beliefs as they fulfil a very human need for certainty in an uncertain world. So whether or not we would be better off without it doesn't answer the real everyday needs of humanity as a whole.
Humanity is made up of a bunch of individuals always, and each individual has varying needs that change over time as circumstances dictate, so there can never be a simple prescriptive answer to the question.

Religious belief--of all stripes--- can be a great comfort in times of need. There is an old saying that there are no atheists in the trenches, ie. when facing one's own possible death.

When my sister (a 'humanist') was slowly dying at a young age, my brother (an atheist) asked me (an agnostic) what I was doing to try to keep going, as it was a harrowing time for everybody. I told him the truth, that I was praying to a god I had never believed in as there was nothing else that I could do.

We are a weird lot, humans, and not half as powerful as we would sometimes like to think.

I think so. I have seen more people damaged horribly by religion than by drugs and alcohol. Uncertainty about life is far less destructive than what can come out of organized religion. Cults, money cons, family conflict, pushing away family & friends and bizarre anger towards society. I have seen how much damage organized religion can do over and over again. Maybe teaching people how to handle uncertainty and the unknown would be better for society than allowing the dark side of religion to run rampant.

Currently in our modern age, the rise of a radical and clearly violent, strain of fundamentalism has overtaken the middle-east. Iran comes to mind.

I would suggest that in this case, like the ancient times of violent christian wars and persecutions, religion being hijacked by monsterous, greedy, power-mad people is the trouble.

In a perfect world there would be limited troubles with religions. Sadly, a perfect world has never and will never exist.

I think we would. We are the only species that seems to have a need to have a belief system, when nature seems to get along just fine without it.