In fact, there is a lack of linear algebra in infinite-dimensional spaces on the general curriculum, and it is an important subject for physics and a few engineering areas besides the relevance for mathematicians. It wasn't clear to me how the dependency between calculus and algebra was supposed to be, but your comment makes it crystal clear.
The discrete transformation is quite fitting for finite-dimensional algebra, and honestly I can't understand how anybody every thought teaching the continuous transformation first and the discrete one never was a good idea. The only explanation is that since everything is thrown on the calculus package without any consideration, there's only time for one, so people kept the most general one.
Infinite-dimensional spaces do indeed come up in e.g. Gaussian processes and the kernel trick. I have to admit I've never really understood them the rigorous form. For ML for example it would be likely more useful than e.g. most matrix decompositions.
I'd guess math uses continuous forms because it's where the mathematical tools are and many things tend to get simpler in mathematical sense when you let something go infinitesimal or infinite. This could maybe be different if digital computers would have been invented before calculus.
I've learned to appreciate that mathematicians think of maths quite differently to engineers or scientists. To them the interest is in the "mathematical objects" and their (provable) properties, not the applications or relationships to "the real world" (and this is probably a good thing in itself) and for engineers/scientists it's the opposite. Maybe something like how linguists vs novelists approach language.
It also comes up a lot in the foundations of RL - the basis of how it is justified (or in some cases proven) to work is contraction mappings and functional operators.
The discrete transformation is quite fitting for finite-dimensional algebra, and honestly I can't understand how anybody every thought teaching the continuous transformation first and the discrete one never was a good idea. The only explanation is that since everything is thrown on the calculus package without any consideration, there's only time for one, so people kept the most general one.