But also on a sphere, every line except for the geodesic lines, I assume are traversed by light, isn't straight, and any sphere, from any dimension, except for the greatest spheres, isn't flat, and thus the earth isn't flat also in your model.
Or do you make Earth the greatest sphere and the light move in strange ways, like in circles through the opposite point to its center?
My model is one in which Earth is round, in flat space with curved space-time (although it could just be in very large spherical space which does actually make more sense in some ways).
But to have Earth be flat you need some form of non-Euclidean space.
Your method appears to work for converting between the Euclidean space Earth is in to some non-Euclidean space where Earth is flat (although I think it would technically need to be inversion in an oblate spheroid or something like that).
The issue is that all the "straight" paths are not actually straight in this space. Light doesn't follow the curvature of this space and instead has to bend.
If you need to appeal to a transformation to Euclidean space such that Earth is a sphere to explain the way light works, that is basically admitting that your model is merely a mathematical transformation and not an accurate description of reality.
It is effectively the same as saying "Well we have Mercator projections of Earth so Earth is flat as these projections are."