It's the first look in here I've had here for several days, J, and I'm not sure I agree with you. For me, a funeral refers purely to what happens on the day on which the body is finally disposed of. This may involve quite a few activities which the undertaker has to arrange.
Perhaps the body has to be take from his Chapel of Rest or the deceased's home to a church or a crematorium chapel. In either place, words - religious or otherwise - will be spoken, hymns sung or whatever and the coffin thereafter will disappear into the depths of the crem or be taken to a place of burial.
But obviously, the undertaker has a lot to do before we get to that stage - much of it best left unsaid! - and the word 'mortician' fits that bill more closely than 'funeral director' to my mind. That's because the 'mort' element, from its Latin sources, clearly refers to handling the dead body. Surely, it is anything but a euphemism.