I'm a few months late to the party, but I think Neal is somewhat off base. This is not Neal's fault, as the reference does not really emphasize the point. Hestenes often repeats the point that vector manifolds need not be embedded. (See, for example, this article.)
After some thought, I think I agree with him. Indeed, the point is almost trivial to see. Often, we take a point $x = x(x^\mu)$ to be a mapping from $\mathbb R^n$ to a manifold $M$. All Hestenes is adding on is that $x$ is a vector in some geometric algebra, as opposed to merely a coordinate tuple. This geometric algebra need not have a higher dimension than the manifold. Rather, assigning this structure automatically makes it such that the tangent space is the base space for the geometric algebra. The basis vectors $e_\mu = \partial_\mu x$ automatically fall out.
Can all points on the manifold be assigned vectors in this way, essentially using vectors in the tangent space to globally cover the manifold? Probably not. But this is no different than how most manifolds require more than one coordinate chart to be covered (see, for example, a sphere).
It is, admittedly, hard to be very very sure of all these things without a comprehensive text on vector manifold theory to prove all the results you would want, but it seems to work pretty well. Hestenes certainly claims that any manifold is isomorphic to a vector manifold, and I absolutely find the usual notion that tangent vectors are $\partial_\mu$ to be little more than voodoo. At the least, Hestenes is right to point out that doing so makes it impossible to use the algebraic structure of GA without tacking it back on. GA is very powerful, so such a loss is quite significant.
I do think Hestenes overplays how coordinate free vector manifolds are. Most important results in differential geometry are proven to be independent of coordinate chart. Rather, I think Hestenes is trying to say that in differential geometry we often say we're working in some arbitrary chart (even if we don't choose it) to do abstract calculations while vector manifolds allow you to cut out that step and not even talk about a chart at all.
At any rate, geometric algebra and geometric calculus are very powerful frameworks. Hestenes' definition of the vector derivative the limit of a surface integral as the volume shrinks to zero is, to me, particularly ingenious, and that requires no reference to embedding at all.