Basically without pasting any non existant (non intrinsic) structure on an actual space, which for euclidian geometry is an euclidian affine space of points. .

The way they did geometry from the ancient Greeks to Descartes.

Coordinates and their maps are the foundation of standard differential geometry. The theory is coordinate free, but riddled with non geometric objects, and with the need to prove that geometrical objects are not just coordinate nonsense.

I am looking for a theory including differential operators that builds directly on the pre Descartes ways to geometry.

Newton developed the entire principia mathematica this way, and I believe he could have used calculus with that geometric approach.

Is there any such exposition that would deal with differential operators like like covariant derivative, vector fields and differential forms, without assuming any analytical (coordinate) geometry