I don't know a recommendation to make; probably it makes more sense to find a field your interested in (pretty much any STEM field will do), and learn the more math heavy version of that. Inevitably linear algebra will come up, giving some motivation for the pure theory.
As for why it comes up so much, it concerns itself with solving systems that look like y = Ax + b, where the Ax term works similarly to multiplication in 1-D. The point is these simple equations are ones we can actually understand! Everything else is too hard.
But there's a trick we have for everything else: if you have some y = f(x) where f is super complicated, you can differentiate. The derivative of f at a point x_0 is the best linear approximation to f. i.e. f'(x_0) is the best matrix A such that y ~= Ax + b near x_0. Now your problem is linear and you can understand it (locally)! Then you can integrate your local solutions into a global one.
The purpose of dot products is that they let you talk about things like angles, lengths, and projections. The point is you learn how it works for arrows and shadows and stuff, and figure out some equations that hopefully make intuitive sense in 2- and 3-D, and then it turns out those equations work in higher (even infinite) dimensions too.
Projections are useful because they let you break vectors down and build them back up, and hopefully the broken down version is easier to understand. Understanding projections in high or infinite dimensions gives some intuition for things like the Fourier transform, where you project a function onto simpler waves, maybe study how a system reacts to those waves, and then use that description to build back how up the system reacts to your original function.
Angles give one way to measure closeness. If you have some machine learning model that figures out a way to map text into a 50,000 dimensional space, you might be able to do it in a way where two sentences are intuitively similar if they are mostly pointing off in the same directions, so if the angle between them is small.
So tl;dr the idea is you learn some geometry with arrows and all that, you figure out some equations from that geometry, and then you realize that those equations and that geometric intuition work anytime you have a linear (i.e. f(ax+b) = af(x) + f(b)) system. Calculus gives you ways to turn non-linear problems into linear ones, so you will find examples of linear systems everywhere.