The failure to eject ("stovepipe") is much more easily cleared by the TRB, with a port down roll, than anything else. The old methods of wiping cases clear just didn't work across the range of those malfunctions and the many pistols having them. Much of the time you got a bloody hand lac and a feedway stoppage because you wiped the case back INTO the gun instead of off the gun.
As for bypassing TRB, some places teach this and I can see merit in that approach. The thing that keeps me from fully going there is that TRB fixes more stoppages than not and so you're in a statistical quandary at that point of which is best. My frame of reference for this is teaching folks that are never going to be above a basic level shooter in both competence and interest (cops). My basic routine is to TRB and if that doesn't work, do the feedway stoppage routine or go to BUG (which most won't carry).
From an institutional approach, it's incumbent on armorers, trainers and supervision to make sure guns, magazines, ammo and maintenance are on the tracks at all times, because in my dark secret heart, I doubt most folks with a stoppage are going to do anything other than stare at it and get whacked by the bad guy they're in a fight with. There was a recent success with my old PD program when they had seven officers shooting at a rifle armed suspect in an apartment. Said asshole had already killed one officer with a shot to the neck and was then in a general exchange with the cops. One of them had a fail to fire stoppage (slide didn't close and lock) and quickly cleared it and got back into the fight. That was gratifying, but I doubt it's the norm to be expected.