Could you explain what you think Gibbs style vectors have to do with Banach-Tarski or the axiom of choice?
As an aside, I'd like to emphasise that geometric algebra gives exactly the same physics outcomes as doing the maths with vectors or tensors or whatever else you like. The difference is essentially just notation. Some things look prettier.
Specific to building _intuitions_ for why the Banach-Tarski arises in ZF+AC.
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.
I don't want this to come across as insulting, but this message sounds more like someone trying to sell me something than an objective scientist. You can just say no, there is no link (intuitive or not) between geometric algebra and Banach-Tarski.
To answer some of the other points
1. No physicist is particuarly confused by Euler angles, and gimbal lock is not a problem in physics.
2. I'm not sure I agree that physics even uses computable reals or rationals. I would say in reality we use fuzzy confidence intervals mostly and not exact numbers.
3. Maxwell's equations look different when written in geometric algebra style, you get one neat looking equation rather than the traditional 4, but its just a difference of notation, the same stuff is happening just written in a slightly different way.
4. A tensor can represent a spinor if you let it transform under the correct transformation rule. A basic spinor just looks like (https://en.wikipedia.org/wiki/Dirac_spinor) a complex vector which you let the Clifford algebra act on.
More generally everything I've seen from Geometric Algebra enthusiasts is just a weird way of doing fairly standard stuff in special cases of Clifford algebras in slightly weird old-fashioned notation. Pretty much everyone I've seen doing real work just does stuff in the Clifford algebra.
Lie groups and geometric algebra remove a lot of problems.
It also applies to differential calculus and ML methods like back propagation and gradient decent.
Gibbs style vectors and the cross are convenient as they tend to match our visual intuitions.
But lots of the 'physics isn't real math' claims just don't understand how the algebra arises from the system.