Perhaps the distance requirements reflect airplane wakes, rather than collision avoidance [0]. Analagously, I don't like to follow cars closely on wet roads, given the plume of side spray [1].
Wake turbulence is mostly a concern for small planes following large planes. In the right conditions (pilots visually maintaining separation), planes do get quite close and at many airports like SFO, it's the only way to accommodate all the scheduled flights: https://www.youtube.com/watch?v=NLCcxCdrL5w
[0] https://www.faa.gov/air_traffic/publications/atpubs/aim_html... subsection 7-4-9, titled Air Traffic Wake Turbulence Separations
[1] https://www.youtube.com/watch?v=C-1-btUcMiA 20 seconds long