No they really aren't. Determinants represent the change in volume in a transformation.
> Ideally [determinants] would not be taught at all, but it's useful for doing exam problems.
This is outright ignorant and wrong. Determinants are fundamental to a myriad of central concepts in calculus and also in engineering fields. For instance, volume integrals or any integral over a parametrization rely on determinants.
In the specific examples you give, 2-forms could be introduced instead to yield the final formula for the Jacobian. I don't think this is an overkill, because it's possible to use 2-forms without precisely proving the theorems etc about them. This is more convenient for math/physics people, and engineers care more about the final formula than about how exactly it was derived.
As another comment here pointed to, the idea that linear algebra is better off without determinants was popularized by Axler - I'm not making this opinion up out of nowhere.
https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differ...
math majors would probably supplement this with additional material