People who can do basic arithmetic on paper have an advantage with back-of-the-envelope calculations and ballpark estimates, and will end up using such simple and effective tools more often. The mental cost of pulling out a calculator and concentrating to enter all digits in sequence is just too high sometimes.
There is also an often overlooked educational advantage in that pupils get early exposure to working with simple algorithms. Becoming used to the idea of following a sequence of unchanging steps, for varying inputs, comes in handy for understanding more complicated algorithms later on (such as those involved in algebra, calculus, and computer programming).
Having an understanding of the meaning of a mathematical concept does not require continuous, ongoing, manual, exercise of its mathematical calculation.
For example, I'm willing to bet that virtually all of you reading this fully grok the concept of "square root". Your understanding of this is probably strong enough that you can wield it in day-to-day usage, successfully employing it to build back-of-the-envelope estimate, etc. Yet how many of you actually know how to calculate a square root by hand?
I've been shown how to do it, but have never done it myself but for once or twice. I also know how to estimate it via via calculus and interpolation. But I don't know how to do it now, and I've not even done it enough times to have gotten an understanding of the workings of "square root" by way of repeated manual application of the mathematics.
So, how to explain the fact that I (and presumably you) are so comfortable in employing the concept?
Knowing how to do a square root efficiently by hand is probably a bit overkill, although it can still be useful sometimes. I was rather arguing about addition, division etc., which come up much more often in back-of-the-envelope calculations; and which also provide a much easier and earlier introduction to the kind of algorithmic thinking that is required in algebra, programming etc.
There is also an often overlooked educational advantage in that pupils get early exposure to working with simple algorithms. Becoming used to the idea of following a sequence of unchanging steps, for varying inputs, comes in handy for understanding more complicated algorithms later on (such as those involved in algebra, calculus, and computer programming).