If you have a displacement vector, and a potential gradient, they can't be the same "kind" of thing (i.e. both vectors), because their dot-product should be preserved. If the slope gets half as long, it's twice as steep.
I'm not sure what two kinds of things you are referring to, but it doesn't seem like they are "geometric objects" and "coordinates". All I'm saying is that you don't need coordinates; they are a convenience, not a necessity. I am not saying you need only one kind of geometric object.
> If you have a displacement vector, and a potential gradient, they can't be the same "kind" of thing
Agreed. You need both vectors and covectors, or more generally "things with upper indexes" and "things with lower indexes". But you don't need coordinates to work with those things. The indexes do not have to represent components. They can represent "slots" (at least that's what Misner, Thorne, and Wheeler call them in their classic GR textbook), in which you can insert vectors (for lower index slots) or covectors (for upper index slots) in order to obtain other geometric objects (and ultimately numbers, which are what you compare with actual measurements).
I agree that we don't need coordinates. Things are things.
But what always got me about the geometric algebra stuff was that they used bivectors for areas, which seems like the wrong thing. If you're integrating a vector field over it, you want a 2-form, not a bivector. I suppose the distinction doesn't matter as long as you're just in Euclidean space, but even then, if you need to drop down to coordinates and do the actual integral, you're still going to want to have changes of variables that work. That leaves me with a kernel of doubt that they're doing the right thing.