> The examples listed feel contrived to get the best case results rather than the worst case.
They're cherry-picked. For addition, it only "makes sense visually" the way the article says it does if the answer lies within the sub-base-5 digit (i.e. the answer is, worst case, less than 5 numbers away).
There's also arbitrary rules in the so-called "easy visual arithmetic" - for some divisions (not all), some strokes have to be rotated. For the long division example, the visual indication of the remainder is reversed - i.e. it's a mirror image of the actual digit.
While I like the idea (the base-20 with sub-bases-5 makes counting easier, and having sub-bases means less memory overhead in memorising all 20 digits), the article itself is spinning wildly to make this seem like "the children came up with it on their own".
The title says "A number system invented by schoolchildren", while the article says that this was the result of a teacher-lead class project which came up with symbols for an existing numbering system.
Aside: With the exception of zero this numbering system is only slightly different from roman numerals - use the number of strokes and the special symbol to determine what number you are at. Counting is easier, and simple addition/subtraction/division is easier with roman numerals as well, but as soon as you need to do common things (approximate VAT for any figure[1]) then base-10 is so much easier.
For really easy arithmetic, using a base-12 counting system is even better (hence, the rise and popularity of imperial measures, which layers a base-12 system on top of base-10).
[1] VAT is 15% where I am, so mentally approximating VAT of $FOO is "10% of $FOO + 1/2 of 10% of $FOO). When it was 14% it was just as easy, do the above and remove 1%.
I think the most interesting checks is with exponentials though? How does this represent e? Pi? Complex number rotations?