What is I think getting lost here is whether there is any real utility in teaching students how to find the inverse of a function.
Option A: There is broad class of functions for which an algorithm for finding there inverses is known, has been safely recorded for posterity, and has been rendered into freely available software. Assuming this class of functions meets the practical needs of 99% of the educated public, we should not torture kids with learning how to find inverses by hand (just as we should not torture them with learning how to add, subtract, multiply, and divide when it would be much easier to teach them how to use a calculator). The true purpose of modern education in math should be to teach how to determine which computer tools apply to problems. That is, the only thing to teach are word problems.
Option B: We should teach mechanics because, done properly, it helps the student understand when a technique is appropriate. But that implies that we should teach the mechanics in a way that meets that goal.
Option C: We teach the mechanics of math to teach disciplined, careful, logical thought.
I admit the plausibility of Option A, but feel uncomfortable with it for reasons I can’t quite articulate. I‘d reject Option C if there are other, more productive, ways to teach mental discipline in fields that computers cannot yet do. With respect to Option B, I’d adopt it if I thought we taught mechanics in a way that led to understanding. I doubt that saying “swap x and y in a formula and solve for y“ teaches what an inverse does and when it is relevant.