In that, notation is really no different from a specialized language. A sentence like "a monad is a monoid in the category of endofunctors" doesn't make sense if you don't know what those strange words mean; but if you understand the concepts, you probably also know what they are called.
Contrived examples of ambiguous expressions like in the article are pointless, since there are always non-ambiguous ways to write the same thing.
The flexibility of math notation makes it flow much better, you can vary the level of verbosity to suit the context. There are even conventions around variables names, so that no one would misunderstand an expression such as f = x^2 and think that f is a constant.
It's a beautiful, intuitive, highly efficient system evolved over millennia. It might not be perfect but if someone had come up with an improvement it would have been implemented already.
I mean, I get that there is a lot to be confused by. There is a lot going on. But the points being argued here seem irrelevant to the point of someone just trying to get pedantry points.
If anything, I think everyone should be encouraged to create their own notation and have fun with it. A large part of the challenge will be translating it between their notation and a common one.
You could substitute quite a lot of other things for 'mathematicians/math' and this would ring equally true. I've noticed that people often defend the system they learned within even (or perhaps because?) when its limitations are creating a barrier for potential new entrants.