Why are doctors ignoring the fact that a nasal septum can be straightened by snapping it into place using just the thumbs?

In some cases where the nose was acutely injured, this technique you describe to straighten the septum may be possible. However, if the broken cartilage and/or bone has been allowed to heal, additional maneuvers may be needed to effectively correct the displacement. 

The septum is 4-5 inches long from the nostrils to the back of the nasopharynx. It is composed of 2 bones and the front part his cartilage. The majority of deviations in the septum  occur at the junction between the bone and the cartilage which is approximately 2 inches to 3 inches in the back of the nostril. If there is a small caudal septal deviation of cartilage on the front part of the nostril, manual reduction with your thumb is possible if the septum itself is straight and not warped, just straight and displaced. On rare occasions we have done a manual reduction to straighten the septum anteriorly. For more diagrams and information, please see the link below

What you say can be done if performed early after injury but the longer you wait the more the nose may need to be operated on to get the best result possibleDr Corbin

The person trying to set a broken nose/septum themselves could do further damage to the nose or adjacent structures without proper instruments or anesthesia.  Donald R. Nunn MD  Atlanta Plastic Surgeon.