No.
Consider that:
Verbal (V)
...the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion.
If you can argue that this is ALL that is needed to cast a purely verbal spell, then anyone or anything with a mouth can cast it.
Let's get more basic: suppose you don't have spell slots left to cast Healing Word. Will speaking the words that produce "the particular combination of sounds, with specific pitch and resonance" set the magic in motion?
It all boils down to the power to cast it, and I'm afraid your familiar does not have it.
Let's look more closely, then, at:
Voice of the Chain Master
Prerequisite: Pact of the Chain feature
You can communicate telepathically with your familiar and perceive through your familiar’s senses as long as you are on the same plane of existence. Additionally, while perceiving through your familiar’s senses, you can also speak through your familiar in your own voice, even if your familiar is normally incapable of speech.
There is nothing in there that ever mentioned -even vaguely- casting of any sort.
The argument of specific-beats-general here is also invalid because "specific" also denotes "explicit" and there is nothing here that explicitly and specifically grants you the power to cast verbal spells through the familiar. This case is more of a general-beats-vague.
Finally, the Find Familiar spell states the times when a familiar can (sort of) cast a spell:
Finally, when you Cast a Spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.
Voice of the Chain Master lacks similar text that lets you cast verbal spells through it, so the feature can't let you do that.
No.
PHB, pg 203
If you can't provide one or more of a spell's components, you are unable to cast the spell.
When you cast Magic Mouth you are embedding a message in a usually mundane object to be spoken at another time. It is physically the object that speaks the words. Since you are not providing the spell's components, you are unable to cast the spell.
Best Answer
RAW: No, the one-word command of command is not the verbal component of that spell.
1. With reference to Verbal components in general:
So, the spell command requires the chanting of "mystic words". This has a couple of implications:
2. Furthermore, regarding command specifically:
So the command in this spell must be intelligible to the target. "Mystic words" implies that to the vast majority of hearers, the verbal component of a spell is not intelligible. So, if people had to understand the mystic words in order to be affected by the spell, it would render it almost useless.
3. Support from rulings on similar spells:
For further evidence of this ruling, see this similar question regarding the suggestion spell - and specifically this text from the Sage Advice Compendium (p. 19) about suggestion:
The suggestion spell's suggestion is not part of its verbal components, so it makes sense for the same to be true of the command spell's command.
However, could you homebrew it otherwise?
While it isn't RAW, a DM could allow the one-word command from the command spell to function as that spell's verbal component in their game.
As the "mystic words" are normally the only component of this spell (it doesn't have somatic or material components), a DM who removes these mystic words might find that doing so complicates the question of whether or not an observer would be able to perceive that a spell has even been cast. If they rule that casting command is now imperceptible to an observer (its only component now being a perfectly intelligible phrase), then that would represent a strong buff to this spell, under the right conditions. Unless ruthlessly exploited, however, even this is unlikely to be game-breaking (though it could be an interesting tool to give a BBEG).
In general, though, spell components are there to help balance spellcasting, in terms of action economy and cost. So, extrapolating this exception and applying this principle to other spells (amalgamating or removing components), as a consequence of this ruling, could cause wider balance issues long-term.