By far the most moral people I have known in my life have been atheist and the least moral have been theists. Of course that is partly a matter of exposure
but it is also because of the fact that theists are so much more prevalent.
As you pointed out Atheists do what is right because it is right and not because they expect some sort of reward or punishment, that is the foundation of true morality.
The idea that atheists believe everything is random is false. There are patterns in nature from rocks to men and there are determining factors that shape the world.
They accept math and science over superstition because of sound logic.
Atheists have no problem accepting that we don't know everything and have no need to make up answers that aren't there attributing them to some unknown, unseen being.
We know that with time we will find the answers if we apply sound reasoning and science, though it may well not happen in our lifetimes.
That one thing above others is what makes them both more mortal and more moral. It is better, more moral, to say we don't yet know than to make things up and then write nonsensical stories to support false reasoning.
I see some who desperately cling to the idea that atheists are closet believers. But this is just another example of people who's position depends on such beliefs.
As to who has been more moral through history it is pretty easy to see that far more war, pain and suffering has been brought by false beliefs. Whether you want to look at the Christian wars, Muslim wars, the evils of Popes like Pope Innocent, or the Salem Witch trials, the Inquisitions, Hitler (Yes; Hitler was a Christian as were most of the Nazis and he murdered atheists along with the Jews and many others.) the list is endless.
There have been some true monsters who were atheists like Stalin, Mao or Pol Pot but these are monsters who happen to also be atheists. They didn't do anything in the name of atheism. While the theistic monsters did most of their evil in the name of their gods.