“The PS1 had no such depth buffer and it was up to the developer to draw all the necessary primitive shapes in the correct order, starting at the back and drawing closer things later.”
The wobbly polygons and warped textures that define PS1 games were not an art style. They were missing hardware. No depth buffer, vertices snapped to whole pixels and textures mapped without perspective. Developers shipped classics around those limits. Now people rebuild the flaws on purpose.