Just to elaborate a bit more. When we speak of the invariance (not constancy!) of the speed of light, what this physically means is that the outcome of experiments is always the same in all inertial frames, ie uniform relative motion has no bearing on the outcome of experiments. This has nothing much to do with units or numerical values.
Yes, it is always possible to describe the same physical situation in terms of different “geometries”, if you so will. You can eg forego any reference to curvature completely by choosing a different connection on your spacetime - the geometry is now curvature-flat, and instead contains all information about gravity in the form of torsion. But all this is saying is that one can draw different types of maps over the same territory, like having a topographical map vs a road map over the same region. That way you emphasise different information, but the actual experience of physically crossing that terrain is always the same, irrespective of what map you use to navigate. This is not revolutionary or mysterious, and reveals nothing new about the world. It’s “kind of trivial” as the poster in your screenshot correctly said.
So I think if you put enough thought into it, it may perhaps be possible to come up with a mathematical description of spacetime in which c is explicitly a function of something. The reason why no one uses such a description is that any measurements of space and time obtained from this description won’t directly correspond to what clocks and rulers physically measure in the real world - you’d have to first map them into real-world measurements, which means additional work and complications without any discernible benefit. Irrespective of what description you use, the outcome of experiments will still be the same in all inertial frames, and this is what we actually observe in the real world.