There is no high principle other than the agreed upon definition of what a function is.

Of course the function that gives you inverse images is nothing more than an inverse function that is allowed to give multiple values.

You can’t use the word function for something that isn’t a function. It makes no sense to say inverse function that is allowed to give multiple values. Hence the need for terms like pre-image of a set.

In all contexts it is important that a function yield a single value for a given input because that is the definition of the word.