This is a non-sequitor. Why would a non-acceptance of the Almighty lead to a life without purpose or meaning? Many of the atheists I've known over the years (and a few of them I knew back in junior high and high school) were not the nihilistic people you paint them to be. Indeed, they'd create their own purposes, such as volunteering for charity work, and set their own goals.
Now, the most "do-nothing" crowd I've ever run with are the "Apocolyptic Christians." These are typically fundamentalists who believe "God is coming soon." Therefore, they don't actually accomplish anything because God'll be here before they finish. My wife's grandmother, a staunch Seventh Day Adventist was of this bent. She tried to discourage my wife from going to college as it would be a waste of time because God would come before my wife finished. Fifteen years later my wife is completing her Master's Degree in psychology having realized that God may not be coming nearly as soon as her (now-deceased) grandmother claimed.
So it's not a non-sequitor. It's a logical conclusion in a culture that has tried (albeit unsuccessfully) to take God out of it.