A sphere is a 2 parameter surface, so it will need some sort of “coordinates”.
A circle admits an easy spinor exponential parametrisation, but I’m not sure how I would pull that of for a sphere, maybe by using two orthogonal rotation planes.

The main purpose would be to be able to solve euler Lagrange equation, with appealing to little to no arbitrary data in coordinates or basis.

Personally I like parametrizing a sphere by the stereographic projection to the whole plane (plus one point at infinity). But you can alternately take any other map projection you like. See Wikipedia: List of map projections for some possibilities.