I'm vaguely familiar with Axler, but I read the beginning of Halmos (finite dimensional vector spaces)
What I meant is that on a pure mathematical point of view they are seen as inelegant as they require a basis to be defined. I don't have a problem with that when you are doing maths on manifolds etc..
From a practical perspective, a lot of engineering problems go from 1 to 4 dimensions and are basis dependents. And the determinant becomes a useful tool.
I agree with you, that in practical paper computations (e.g. on exams) that determinants are an indespensible practical tool.