I’d like to create a formula that takes input from a table to determine it’s operators. So for instance, instead of saying X + Y it would say X thisrow.operator Y and a user could then easily change the operator to + - / * = > etc, by just changing that cell and not needing to know how the formula works. This is a simplified example for the purposes of the question.

I know this is not possible the way I have laid it out, but is it possible at all?

What you could do is the following:
In the formula column, write a (series?) of if statements:
If thisrow.operator = “+”
then X+Y,
If thisrow.operator = ""
then XY,