I read your whole article. I respect and enjoy your writing, though I disagree with you (respectfully) on several of your points here.
For now, I'll simply say...
There are a few different angles to take the question that forms the title. Of course, people can do good stuff without believing in God or subscribing to a particular religious belief system. That's obvious.
It's much harder to pull that off, however, the bigger the group gets. Or the bigger the nation gets. In other words, if you have 10 people doing their own thing, that is more manageable than 10 million people doing their own thing -- or 100 million or 300 million.
And that's why I would (respectfully) go with George Washington on this question instead of the arguments you are making. In fact, I would say just about all the founders agreed with Washington on this.
Finally, you have the ontological question of how to define "good" and source "good." That definitely leads you into some considerations of which I don't believe you extend enough attention or respect.
Nonetheless...
I appreciate your writing. It makes us all think. And that's a good thing.