More or less agreed. I think though that one reason the geometric product is so tempting is that if you take matrix representations of all of these objects, then the geometric product is literally just straightforward matrix multiplication.

Because of that, it just becomes so tempting to try and phrase everything you can in terms of this geometric product. I'm very sympathetic to the temptation, and I even think the geometric product has some great uses (it shows up a lot in some physics I do), and using it makes writing rotations a treat, but I think it's still vastly overemphasized by GA people.

I still don't really know what my favoured notation for differential geometry is, I find myself switching around so much.