The spirit of the Army version of the rules is that if there is more than just two folks playing, no additional witness is required. We've run several Battletech tournaments in brackets as you describe. But since every player and the referees are all TRMN members, even though each individual bout is 1v1, we don't have an additional witness because the whole room is full of TRMN members playing in the tournament.
Taking off my official hat for a moment and offering a personal opinion: I trust my Dragoons not to make up game sessions. We Drill twice a month with each Drill having 5 gaming hours, and in addition most members are also in many other games throughout the month. There's no need to cheat or fake gaming hours. I'm not sure there would be rampant cheatery in TRMN if there were no witness rule. I mean, one would hope the second person involved in the game report would keep the first honest. That said, that is not how the rules are written, and in my official capacity as Army Marksmanship Director, I do my best to explain and enforce the rules as they exist.
JEFFREY A. WEBB, CS, MG
Officer Commanding, 37th RCT "The Royal Dragoon Guards"
First Baron New Caladan