I'm not talking about linear algebra as a field of mathematics in some kind of ideal sense here. Yes, obviously, it's a foundation for geometric algebra. You don't need to convince me of that.

However, elementary linear algebra classes don't teach you about linear operators, they teach you about things like matrixes and cross products. In these basic classes, a "vector" is a "thing with X, Y, and Z coordinates". So when you get to physics, you use the cross product to write a formula for magnetic field. You have to remember that the magnetic field is transformed differently from other vectors according to some special rules. And engineers call this stuff "linear algebra". Mathematicians agree that it's linear algebra, but we know that there's a lot more to linear algebra that goes beyond that.

Alternatively, they could calculate the magnetic field using geometric algebra, and express it as a bivector, at which point all of those special rules vanish.

That's why Hestenes's book is called "New Foundations for Classical Mechanics". It's not that linear algebra is not the foundation for geometric algebra. It's that classes taught in colleges which are called "linear algebra" teach you the concepts used by Gibbs and Wilson in the book Vector Analysis, and these concepts don't generalize to different numbers of dimensions. GA does. Maybe the problem here is that we don't have a special name for that field of study which uses cross products, if had a different name for that stuff, say "vector analysis" after the book first appeared in, we wouldn't have a problems saying that "geometric algebra is an alternative to vector analysis".

GA is a nice alternative to the stuff they teach engineers scientists under the "linear algebra" banner.

Another example… look at Stokes' Theorem. The version with differential forms is a nice alternative to the version with just a cross product.