Can you change the perspective in POV-Ray, so that convergence between parallel lines does not look so steep?
E.g. change this angle (the convergence of the checkered floor into the distance) here
To an angle like this
I want it to seem like you're looking at something nearby, so with a smaller angle of convergence in parallel lines.
To illustrate it more: instead of a view like this
Use a view like this
You need to reduce the field of view of your camera's view frustum. The larger the field of view, the more stuff you're trying to squeeze into the output of your camera's render and so they parallel lines will converge faster. So in your first example with a cube, the camera will be more focused on the cube and the areas just immediately around it, than the whole environment.
The other option is to make your far plane much closer to your near plane, so you don't see many things that are far off. So in you first image example, you'll only see the first four or five grids instead.