It is not an issue until you start trying to teach any sort of fix point theorem, at which point the wheels fall off because the notion can't deal with higher order functions at all.
I guess I just don't get your point. D 2 = 0 doesn't imply anything, it's just shorthand to make working on something less tedious for people who are already familiar with it. It seems perfectly expressive and sane: mathematicians have successfully studied differentiation using standard math notation. 2 and (λ (x) 2) aren't 'treated the same', that's just a strawman.
Funny how its also used in every textbook for people who aren't. This is getting to be apologia on the levels that I'd only seen in history books about why Roman numerals were superior.
The premise of your argument is that this notation is confusing people, or it's impossible to make explicit what it means. I don't buy it, you even explained the type elision using standard mathematical notaion in your original comment.
So if there is some superior notation that would benefit all of maths, that sounds very interesting but I don't think you've made that point very well. Maybe a better example is needed, because on its own, eliding repetitive information from notation does not seem like a problem.
Because I'm not going to write a few dozen pages of latex in a text comment. If you're interested thrown me $2k over bitcoin and I'll spend the 8 hours it would take to make this rigorous at below my usual rates.
