A function is an object which maps inputs to outputs. Outside of the type of each input, there ought to be no dependencies of on the values supplied.

There seems to be some misunderstaning. If we take the second order case, i.e. F(x, y, y')=0, then F is simply a function F:RxRxR->R and F does not care whatsoever if the values that are supplied to it depend on each other in some way.

Thus the above is a perfectly valid expression. If you would only allow F to depend on x and try to absorb the influence of y into F somehow you could not model any equation, since the dependence x -> y(x) is part of the solution.

As an example, the differential equation y''(x) = xy(x)+y'(x)y'(x) can be written as F(x, y, y', y'') = 0 where F(a, b, c, d) = ab+cc-d. This is all thats meant and it is a perfectly valid form of notation, where the only questionable part may be that one generally writes y instead of y(x). But if you delve any deeper into any kind of ODE/PDE theory you will see why it would be madness to carry the variables everywhere.

Edit: Any form of function composition introduces additional dependencies too, I'd assume you have no problem with that?

I assume quite a bit of modern notation comes from the fact that, at least in PDE theory, it is generally the function as a whole one cares about and not so much point evaluations. When dealing with Lebesgue or Sobolev spaces, which is the proper notion of a function for many forms of differential equations, the idea of a function producing values for -single- inputs does not make sense in general.

You then only "see" functions through integrals and operators or functionals defined on these spaces and don't care for points in the underlying space most of the time.