I have a split mind on this. In Chaz's case, clearly its not an issue how he chooses to do his math, but for students, calculators are a crutch. It can be very very useful to look at an equation and immediately see limits (like if this variable goes to 2/5 then this variable goes to 0.) If you are comfortable with your multiplication tables as well as understanding roughly how logs, exponentials, etc look you can do this easily, but if you have to use a calculator for everything you won't be able to see it.
A more basic example of this would be factoring equations. I can look at a quadratic polynomial and pretty quickly give someone the roots. With a little more practice I can do the complex roots without having to do the quadratic equation. Without the quadratic equation factoring roots isn't something that a calculator helps you do, its more an intuitive level of comfort with numbers.
That's really what's important. That comfort with numbers. And that's why it doesn't matter so much with Chaz. We can assume that his education and work experience up to this point gave him that, but with students you can't assume that, which is why its important for them to not rely on calculators. There is a lot of value in seeing a curve and saying, hmmmmm.....that looks like a natural log (there was a situation at work where someone realized that the curve for something was based on the michaelis-mentin equation, which was a crazy catch because this was not a biological system. Only with his comfort with numbers was he able to catch that similarity.)
Part of my attitude for this came from my Freshman Chemistry professor. He told us in the beginning that we would not be allowed to use calculators on the tests, and that there would be a couple of problems that required us to do some relatively complex math without them (including natural logs, he showed us how to interpolate a rough estimate of the latter pretty quickly), and if we got within 10% we got it right. As he put it, getting within 10% can win you a nobel prize.