I think in your logic you still get 0 + 0 = 1. Look, we are summing up two zeroes, so we are counting them: 0, 1. Hence, the result is 1, not 0. And if we add one more zero we will get 2.

So, I think this part is not quite correct:

And you're right, if you put 0 dollars on a table and then add 0 more dollars you wont get 1 magic dollar out of it. But don't forget that we're not done yet, we still need to 'translate' that result of 1 into it's Nth digit. In this case the 1st digit is 0. So 0 + 0 = 0 still, even when you start counting from 0.

You applied an extra translation step.