I agree with much of your thinking, but I'm a lot more optimistic.
Religion can keep people from seeking a better life in the here and now. Here's an extreme example: Christianity was drummed into African slaves around the world, at least partly to give them a better existence to look forward to that did not at any stage involve killing their masters.
However, look at what happened in the end. While there were a few slave uprisings, that's not what freed them as a whole. Freed slaves and their descendants slowly gathered the political, financial and social impetus to change basic attitudes toward slavery worldwide, until they had enough public support for such achievements as the Emancipation Proclamation. The cause wasn't only championed by believers; William Lloyd Garrison, Ralph Waldo Emerson and Robert Ingersoll were all atheists or agnostics, and Abraham Lincoln thought pretty darn freely even if he was finally a believer.
To answer your question directly:
I'm not confident that there will ever be a proof of the nonexistence of gods. It's the ultimate unprovable negative. If it did happen, however, I see two major reasons why we would likely avoid anarchy:
Firstly, pragmatism. As the slaves would have realised, violently subjugating others and/or completely destroying the society in which you live is probably not the best way to improve your lot. Peaceful advancement from wherever you are will win you the respect of your peers and any observers, rather than widespread condemnation and retaliation.
Secondly, empathy. After you realise you have only one life to live, if you care for your fellow humans at all you extend this epiphany, and it dawns that they each only have one life themselves. Finding ways to be happier without harming others will reward whatever conscience you have.
Finally, C510, don't take this the wrong way but consider that your question is rather patronising towards its subjects. The idea that while you and I do fine without faith, "people" deprived of it would go off the rails is somewhat elitist in itself.
- SmartLX
It's because we actually have little desire to continue evolving as we have.
Natural selection is a reality in that it's the way things have happened so far. It's not an ideal, and it's not something to strive for. Many biologists like Richard Dawkins openly call for people to defy their Darwinian instincts.
Species that evolve become better suited to their surroundings. Humans have now developed the ability to change their surroundings completely, so evolution no longer serves the same purpose. It's still happening, but we don't know where it's taking us. Genetic research may allow us to take the reins and drive it ourselves.
I don't think humanity is being degraded by simply making those with various problems equal in society. It just means their issues aren't as important to us anymore. Besides, natural selection is still at play because now that such people have a level playing field, it's up to them to overcome their disabilities and procreate, and they'll self-evidently find that more difficult.
- SmartLX