I'm looking something akin to a synthetic approach, but augmented with formulas that can be used without adopting coordinate systems

Yes that's what's he's doing, but I don't see how that's different than an embedding into an Euclidean space. Embedding is still embedding. When you talk about standard charts and coordinates manifolds there is no embedding whatsoever. So a priori his approach can't be an alternative to the standard way no matter how clumsy he says the standard coordinate approach is.

You just said in a comment some people say this is the alternative way to do differential-geometry, using vector manifolds instead of pseudo riemann metrics. If hestenes doesn't have a precise definition of what he means who has?

@ben mckey if vector manifolds are not intrinsic than I don't consider them a coordinate free alternative to atlases and charts formulation of manifolds.

I don't think he does that. I think I might have a pretty good idea now what he means. But would still like it to be a it more precise.