Determinants are sometimes useful for symbolic inversion of matrices, but are not used for numerical inversion of non-trivial matrices (for reasons of both performance and stability).
It has been a long time since I studied functional analysis, and I was never a physicist of any sort, but my limited understanding is that there are two common generalizations for the infinite case, both of which handle the cases of most interest to physics but are not available for general operators, and which mostly agree up to a constant factor, or are at least capable of producing compatible results in some sense. I'm not sure I'd say they "nicely generalize", though I'd admit to being hasty in saying that they don't generalize.
And yes, I agree that they're widely used in physics (which is why I think they should be taught to math/physics majors).
It has been a long time since I studied functional analysis, and I was never a physicist of any sort, but my limited understanding is that there are two common generalizations for the infinite case, both of which handle the cases of most interest to physics but are not available for general operators, and which mostly agree up to a constant factor, or are at least capable of producing compatible results in some sense. I'm not sure I'd say they "nicely generalize", though I'd admit to being hasty in saying that they don't generalize.
And yes, I agree that they're widely used in physics (which is why I think they should be taught to math/physics majors).