A tensor is a differential structure with pointwise linear properties. But the important thing is not the pointwise behavior but the differential nature of the “family of tensors in non-Euclidean space”.

This is too strong. There are important fields whose points are not tensors (e.g. spinor fields) and treating them like tensor fields risks introducing all sorts of subtle confusions.

Of course you are right but I was just trying to explain to OP that certainly General Relativity is not just “linear algebra”.

That's very weak