But even in high school topics start to talk about functional equations (calculus, e/ln). I'm not sure the <function> vs <value> doesn't come from the mainstream imperative paradigms and only that.
The distinction isn't between functions and values in general, it's between the function being called and the arguments passed to the function being called. The difference isn't in the things themselves, it's in the role that they play in the specific expression we're reading.
This argument is rather strange. Maybe for people who never interacted with different -fix notations ? Human language binds concepts with arguments in all kinds of direction .. I'd be surprised this is enough to annoy people.
Natural language isn’t precise and a lot is inferred from context. Exact order does often not really matter.
In formal languages, however, you want to be as unambiguous and exact as possible, so it makes sense to use syntax and symbols to emphasize when elements differ in kind.
Incidentally, that’s also why we use syntax highlighting. One could, of course, use syntax highlighting instead of symbols to indicate the difference between function and arguments (between operation and operands), but that would interfere with the use of syntax highlighting for token categories (e.g. for literals of different types).
We also alternate between various positions in programming languages, so it's not a matter of formal precision.
Not sure what you mean by “alternate between various positions”.
Of course, theoretically you could also view the function as a parameter of the computation and/or the arguments as specifying an operation (in particular if those are also functions), but for most concrete function invocation that's not generally the mental model. E.g. in "sin(x)" one usually has in mind to compute the sine, and x is the input for that computation. One doesn't think "I want to do x, and `sin` is the input of that operation I want to do". One also doesn't think "I want to do computation, and `sin` and `x` are inputs for that computation". It's why you may have mentally a sine graph ranging over different x values, but you don't imagine an x graph ranging over different functions you could apply to x.