This is not what I, and I would posit most physicists, understand as a physical treatment of what general covariance is in physics. General covariance is that the equations look the same in any coordinate frame - any meaning that the transformations can be any function. The only limitation is that the functions be differentiable, maybe n or infinite times (diffeomorphisms).
Newtonian mechanics has the Galilean group as its definition of covariance. It means transformations to inertial frames only. That is not general covariance in the way physics either works or is understood.
Possibly you can define mathematical contraptions that allows something more general, but it would be a mathematical technique, not a deep property of the physics. The Lagrange or Hamilton equations (which yes are more than contraptions but still no newer physics than Newtonian mechanics) may look the same if you change the p and q's to another coordinate system, but the equations of motion are different in non-inertial frames. The centrifugal and Coriolis forces are not real forces but appear in rotating coordinate frames. And yes, you do have relativity covariant (or Lorentz covariant in special relativity) but it is not space, it is 4D spacetime and a +/- 2 signature, and not 3D.
I saw the Wikipedia article, and for physicists also Landau and Lifshitz and Goldstein/etc, on Newtonian covariance and this is consistent. Wikipedia simply calls the Galilean group the covariance group. In physics we call it the symmetry group. In that article that the OP referred to it clearly says covariant for special relativity, but they say'Lorentz covariant'. Newtonian mechanics is similarly 'I Galilean covariant'. BTW, the article referred to by the OP is about philosophy of science, not about science.
Did some new physics that makes Newtonian mechanics generally covariant get discovered more than 200'years after Newton?
Or am I not understanding something you are trying to say? And if so, does it make any difference in physics and is it represented in accepted refereed physics journals?
By the way, in general relativity having the general covariance makes a difference. In radial normal coordinates the Schwarzchild metric has a singularity at the horizon. In Penrose or Kruskal coordinates it is seen that it is not, and in fact you can use them to understand the causal structure of black holes. Many other reasons. It is not for naught.