[eigenket]

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.