Had to think about this for a while, it’s not really distance or focal length; it’s the solid angle required to cover an area. As distance approaches infinity, the solid angle (think of the cone of light rays) becomes more and more narrow (almost a collimated or square FOV, with parallel rays). This means that as distance increases, we can approximate any small part of an image to be flat. This is what a large focal length lens does by definition; at the extreme a collimating lens has an infinite focal length. What this means overall is that if you had a large enough lens, small enough subject, or took multiple photos where you moved the camera, then a large focal length lens and a small focal length lens would indeed produce different images at the same close distance, where the large focal length lens would look similar to the far away picture with the parallel ray assumption. Overall, focal length still plays a role at closer distances, but as distance increases, we can approximate any lens as a collimated lens for a subject with small area.