In my understanding the purpose of using tensor equations in GR is to ensure that they are true in all coordinate systems. I understand that writing equations tensorially ensures this will be the case; however, are there not non-tensor equations that would also be true in all coordinate systems?
For example, one can define a tensor by its components and how they transform from one coordinate system to another (the tensor transformation law). It seems to me that you could define some other quantity that transforms according to another transformation law, and that equations written in this quantity would also be valid in all coordinate systems.
I've also seen tensors defined as geometric objects on the manifold that act as linear forms on the tangent and cotangent spaces on the manifold. This geometric definition immediately guarantees coordinate independence. Again, I don't see why we can't define a more general geometric object (i.e., not a tensor) and make that the basis of our coordinate independent equations.
To summarise, why is there an emphasis on tensor equations in GR when it seems to me that there should be plenty of non-tensor equations that are valid in all coordinate systems as well?
EDIT: As an example, consider some arbitrary mapping from the tangent space to the reals that is not linear in the tangent vectors. This is a coordinate independent definition. The only difference between these objects and tensors is that for tensors, the mapping is linear. I suppose non-linearity means that these objects won't have straightforward, easily interpretable 'components' in each coordinate system, but I don't see why we still couldn't make important statements about the geometry of spacetime using them.