Miss Manners would certainly advise that you shouldn't talk with your mouth full. There's the visual aspects, to be sure, but there's also the issue of garbled speech with the attendant lack of actual communication. Better to have your vocal system free of debris if you want to be heard clearly and be understood.
Articulation -- the clarity of sound -- is also found in music. There are even musical terms to help describe articulation: staccato, marcato, tenuto, slur, accent, sforzando, legato, and others. Each one gives the player an instruction on how the notes are to be played in order to fit the music.
Imagine if the guitar player or the singer routinely played inaccurate or garbled notes. We'd never tolerate bad notes, sloppy chords and a general lack of caring about doing it right. Unfortunately it's all too easy for a drummer to get away with poor articulation.
Sloppy technique helps to cover up a lot of shortcomings. One trick is to use scratch rolls. Rather than a clean, simple fill, you hear a nondescript bzzzzt. There are some music styles where this might work, but usually the rest of the band will be looking for precise rhythmic statements.
If your notes aren't clear, they can't be heard properly, and nobody will be able to tell if you're placing the notes accurately within the metre, and this obscures the time pulse. As long as you end up somewhere close to the beat, it'll do -- at least that seems good enough for some people. Mushy sounding tuning makes this even easier to pull off. You cannot articulate clearly on a drum that's tuned below its useful range.
Articulation is also an important part of your style. You can spot certain drummers by their articulation alone. Buddy Rich and Bill Bruford are easy to pick out just by the way they make a sound.
So make it a personal goal to speak clearly, both off and on the stage.