I think the administration is being lazy and would rather let the Radical Mourner's coworkers cover for her as long as they are willing.
I wonder if they are concerned also about what would happen if they tried to insist she go ahead and teach. Would she just cry and make a big inappropriate fuss in front of her students, and not really teach anyway? It's probably easier (and better for the students) to just let her cry elsewhere.
I still don't think this is a fair situation, though. If she was told to pull herself together, if she did, that would be a great solution to the problem, but if she didn't, the consequences could be a lot worse, and I don't know if she would be fired or if she should be fired or what would happen, but it would be very disruptive.
If it were me, I would be talking to administration about it. I would just keep it strictly factual. I wouldn't say anything about how I thought it was "silly" (even though I think it is), I would just treat it as though she had experienced a legitimate personal loss ( that was affecting her work in this same way. Because even if it were, this problem would still be a problem for OP and the other teachers. So I would try to discuss with administration the possibility of handling this in a smoother way when it comes up.