What I saw was two storied coaches meeting for the first time again in (16) years and one abruptly depart after a menial attempt at pleasantries. I was surprised given that the story before this game was not the Cowboys vs. the Commanders, but rather the re-acquaintance of two HOF coaches.
The players are not going to leave the dressing room within 20 seconds, Gibbs has the time to exchange with Bill and present himself well on primetime television.
It's no big thing, but in life it is the little things that separate the mediocre from the excellent. Stuff like looking a man in the face and greeting him warmly with a handshake if tradition necessitates that.
IMO, if there was any venue that requires that tradition, it is the end of a sporting event. From junior players at elementary school to pro sports you shake your combatants hand and leave the events of the match on the field.
That's why the cameras follow football coaches to the middle of the field.