Cambridge University Press make their books free until May

I disagree. I used Kleppner and Kolenkow as a freshman, and it has pretty much the same content as this book, and I would consider Kleppner and Kolenkow to be an introductory book. Sure, it's more advanced than more common freshmen books like Halliday and Resnick, but it is still very elementary mechanics. Newton's laws is just the tip of the iceberg that is mechanics. Most advanced books, including upper-division undergraduate books, would at least introduce analytical mechanics, whereas this book doesn't. If you want more advanced treatments of mechanics you should look at Landau and Lifshitz, or Arnold, or Marsden, or even Lanczos (the latter three making moderate to extensive use of Riemannian geometry and can be daunting for that reason alone).

> but the key to this system is in essence that you leave any kind of coordinate projections till the very end

That is not the definition of coordinate invariance, at least as defined in physics. Newton's laws aren't coordinate invariant because the acceleration involves the second derivative of coordinate basis vectors, and that is zero (or constant) only in very few coordinate systems, like Cartesian coordinates for instance. Of course, the statement F = ma still holds true, but becomes very cumbersome to use if, say non-orthogonal coordinate systems are used. As a simple example, try finding the equation of motion of a particle constrained on a smooth surface, say a paraboloid z = x^2 + y^2 in Cartesian coordinates using Newton's laws. Of course, this problem can be solved using Newton's laws by introducing Lagrange multipliers, but that essentially amounts to finding the constraint force. But it would be much better if you could avoid finding the constraint force altogether.

> A vector is the direction and magnitude per sé, devoid of the concept of an origin or any particular unit vectors with which you could represent that vector numerically.

You're right that it is important to distinguish between the geometrical meaning of polar vectors and their representations in a particular coordinate system. It's even more confusing when axial-/pseudo-vectors, e.g., the cross product of two polar vectors, are introduced since now they do depend on the handedness of the coordinate system used. And why should nature care about handedness? (It actually does, but that's a story for another day.) And this is precisely why analytical mechanics, which is an inherently geometrical subject, was invented in the first place! All equations in analytical mechanics are scalar equations and look the same irrespective of the coordinates you use to describe the system.