> We can't say that a function equals a set

Why not?

Can we not so easily speak of the set of all inputs and the set of all outputs? Why not exactly then is a function not a set of morphisms/arrows?

To me, x->x+1 and {(x,x+1)|x∈R} seem the same[1]

Because it seems useful to be able to make statements of the cardinality of that set: If there are a lot of rules, then that set is big, but if there are few rules (like x->x+1), that set is small. This is enough to permit some analysis.

It also preserves "plus" for sets, because a function plus a function is the sum of those rules being considered.

What is it do you think I am missing?

[1]: I understand I don't really mean big-R here because computers have limited precision for fadd/add circuits, so if you'd prefer I said something slightly differently there please imagine I did so.