If you think the constraints make it trivial, try producing a solution with the same constraints. (intersection allowed but pinching and cutting not)
I think the constraints make it impossible. It's obvious that allowing free passage through a surface is the same as cutting.
Self-intersection is not the same as cutting. The disconnect you’re experiencing is between the mathematical description of the problem—-does there exist a smooth homotopy of immersions between two embedding a of the 2-sphere in 3-dimensional space—and the colloquial description. The mathematical phenomenon corresponding to cutting would be discontinuity of that homotopy, which is different from the description of self-intersection, which amounts to the difference between an embedding and an immersion.
I'm not sure what sense you're using "the same" in. Eg a cylinder can be transformed into a mobius strip (in 3d space) by cutting and gluing, but if we allow only self-intersection, this can't be done. To me this shows that they're not "the same".
Meaning, you can't pinch or cut. So sphere eversion is obviously impossible...but the suddenly, "I can pass surfaces through one another!"
So I guess the game here is not in finding interesting solutions within a set of constraints (no pinching or cutting!) but in just making up arbitrary new rules when the problem becomes intractable?