In calculus why replace the y coordinate with F(x)?

Because in calculus, we are seeking all kinds of functions.

There are "functions of x" (where there is only one variable -- x)

There are functions of many variables

f(x, y, z) = 3x^2 + 2y - 7/z

is a function whose value depends on three variables.

And so on.

The basic "formula" of the type

y = 3x - 7

is just for beginners.

It is to help you picture the function on a Cartesian (x-y) plot.

The problem is this:

Once you are used to picturing functions on an x-y plotting graph, how do we get you used to "picture" the result of 3-dimension functions in a 4th dimension?

We do that by getting you used to the notation of f(x) instead of y.

For functions of a single variable x, they are interchangeable.

It's just a question of which is more convenient and what you want to emphasize.

It depends on what you want to emphasize. f(x) focuses attention on the RHS as a function of x.

You can say f'(x) = y' = y'(x) = dy/dx.