The more precise statement of "math says rolling more dice makes things less random" is that if you roll ten six-sided dice and add up the answer, the result will be less random (on its scale) than if you merely roll one six-sided die.

Even more precisely: the outcome of 10d6 is 68.7% likely to lie in the range [30,40], while the outcome of 1d6 is only 33.3% likely to lie in the corresponding range [3,4].

I think the quoted portion of the article addresses exactly this point: people were scared of rolling many dice because this meant lots of randomness, but the math says that the opposite effect occurs.

As to your other points (starting with "kind of a slap in the face"), that is addressed in the article, but not the quoted part. In summary: both rolling dice and drawing cards is random, but there's a bunch of reasons why the randomness of drawing cards isn't as frustrating. (It can be frustrating too, though.)