Why should it be in linear algebra? You mean as in doing linear algebra with functions in infinite-dimensional spaces? What I recall mostly (not) learning from linear algebra were various classifications of matrices and their properties (which made very little sense to me at the time too).

I'm not exactly sure where FT should belong in the math syllabus. It's heavily related to trigonometry of course but it is an integral transform, so needs a bit of calculus. Although discrete versions are probably easier to grasp with just multiplications and sums, and there is quite rarely much actual integrating as in find-the-closed-form-antiderivative going on.

Maybe trying to have too "unified" syllabus for engineers/scientists is not the best way in practice. Instead there could be more discipline-specific teaching (something TFA hints to too). With ML (and 3D graphics earlier) something like this seems to be happening to linear algebra. The "mathy" linear algebra (that I studied at least) is mostly about properties and decompositions of (complex) matrices, but in ML/3D engineers very rarely use such things (but do need stuff outside "traditional" linear algebra like tensor products, projective coordinates and rotation groups).