I am aware that functions are intuitively viewed as processes, but I
feel like this intuition is not well captured by the definition in set
theory, given by a relation (maybe its just me but thats the way I
feel about it). It is intuitive that one can have subsets where the
objects satisfy certain properties. However, in this case one has a
set of objects where the 2nd "coordinate" depends on the first, [...]
This lead to the question: Why should collections of pairs be allowed where
the second coordinate depends on the first?
I am not sure that functions are intuitively viewed as processes, or that the intuitive concept of function is linked to the idea of dependence. At least, this is not the intuitive idea of function in current mathematics.
The idea of dependence can be present, nowadays, in applications, when a function is referred to specific objects, and there could be an intuitive idea of 'causality' between variables. Or it can be suggested by functions that are expressed by a formula. But they are, by now, just instinctive suggestions, far away from current mathematics.
In mathematics the idea of ‘dependence’ could be present in the past, when the concept of function wasn't still clear from a mathematical point of view, and mathematics, in this regard, was intertwined with physics or thought of as something that must be described by a formula. There was a debate between 'geometric' concept of function (a 'curve', in the cartesian plane; Descartes, 1637) and an 'analytical’ concept, according to which a function is defined by a formula (Leibnitz, 1692) (in the XVIII° century the two concepts collided in the study of the 'vibrant cord'). It took a century of debate to arrive at a unification and at the modern concept of function, with the Dirichlet's definition (1837).
The definitions usually accepted and the very use of the concept of function in modern mathematics are far from the idea of 'dependence'. No mathematician, nowadays, thinks, instinctively, of dependence if thinking about a function.
The definition of function more widely used in mathematics, currently, is the Dirichlet's definition:
Given two sets A and B, a function is a rule which to each element x in A associates one and only one element y of B.
Where the term 'rule' is left undefined, and the rule can be of any nature, not necessarily a mathematical formula.
In this definition it is already hard to see an idea of dependence/process/causality, as it is far from physical objects, and also the arbitrary nature of the 'rule' is underlined.
And the sets A and B can be any kind of sets, also sets as groups or topological spaces: it is not very natural or intuitive to say that an element of a group or of a topological space 'depends' from another or to envisage a 'process' involving them.
Even if the definition above is the most currently used, the evolution of the concept of function in set-theoretic language, as a subset of a cartesian product, cuts further the link with intuition, but, in my opinion, this cut is already completely present in Dirichlet definition.
The set theoretic definition has the purpose of avoiding the undefined concept of 'rule', and substituting it by the (defined) concept of 'cartesian product', but the abstract nature of the definition is already in Dirichlet formulation.
As an example of what can be the instinctive feeling of a mathematician when speaking of 'functions', we can refer to the famous Dirichlet's function, one of the most famous function of mathematical analysis.
The Dirichlet's function is a function that associates to each real number 1 if the number is rational, and 0 if the number is irrational, so we have a function that jumps infinitely many times from 0 to 1 in every interval.
The second image is a possible approximation of the graph of Dirichlet's function.
It is hard, in front of the Dirichlet's function, to have an instinctive feeling of 'dependence' or 'causality' or 'process', I think.
And also, mathematicians, currently, work with ‘spaces of functions’, very abstract spaces, with functions thought of as objects of these spaces, a very abstract world of concepts.
The idea of dependence or process is completely lost.