Coordinates have their use, and coordinate-free formalism has its use.
This is long understood very deeply by algebraic geometers:
a) An affine scheme in the sense of a solution locus to polynomial equations is manifestly a coordinate-driven space. (The fact that we are talking about polynomials involves the choice of coordinates, namely of their free variables.)
b) By considering "functorial geometry" over the category of affine schemes, one obtaines a very abstract coordinate-free picture of geometry, via "schemes" and "algebraic spaces" and so forth.
When doing geometry, generally we don't want to discard either of these.
There is a differential-geometric version on precisely this picture of algebraic geometry, this goes by the name synthetic differential geometry.
A development of perturbative quantum field theory in these terms is available here:
PhysicsForums Insights: Mathematical Quantum Field Theory
- chapter 1. "Geometry"
(this is all coordinates on Cartesian spaces, analogous to affine schemes);
- chapter 3. "Fields"
(this introduces the abstract coordinate-free functorial geometry induced thereby)