GA gets rid of the external conventions for coordinate and chirality and also uses SU(2) which is simply connected vs SO(3) which is not. Rotors in GA can be used as elements of the algebra like any number avoiding the complexity of Euler angles, gimbal lock, etc....

GA's rotors are geometrically intuitive and can do rotations around an arbitrary axis, where quaternions are limited to an axis through the origin.

As Banach-Tarski is not physically realizable and because physics uses the computable reals, rationals and other aleph naught numbers it doesn't cause a problem there.

Lots of important work resulted _from_ the Banach-Tarski paradox but really it is just a cautionary tail about ZF+AC and on-measurable sets as far as physics goes.

What I was talking about is tools about building intuitions on why it arises.

Note that the maths aren't exactly the same, as an example Maxwells equations require four separate formula to express in Vector Calculus vs just one in GA. I don't think I fully comprehended the connection before learning GA.

This is also digging deep into the implications of your chosen groups and resulting algebra but as an example:

A Tensor can't represent a spinor but an even multivector can. A pure grade multivector can only completely represent antisymmetric tensors.

You can look into Dirac's belt trick as a physical example showing that SO(3) isn't simply connected but it arises in E(3) in that particular case too.

I wish this site had latex support, so I apologize for the above which is probably of little value in reality.