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.

Ask your DM

Unfortunately, there really isn't any RAW for how to handle damaged tongues or even being mute. For the most part, and except for an optional damage rule from the DMG, there really aren't RAW mechanics for PCs losing body parts.

And all of this is dependent on how a DM defines the disability.

If they rule that the character is mute, then they are effectively under the Silence spell. If it is somewhere in between mute and full speech, then they'll have to make a ruling on how that will affect the Bard.

Workarounds

If the DM does rule that the character is either a mute or their speech is damaged enough to affect Verbal components, then there are some remaining options.

Cast Regenerate

This is the most obvious solution. Either have a party member or a hired caster cast Regenerate to regrow that tongue. Once regrown, all is back to normal.

Dip Sorcerer for Subtle Spell

If regenerating is not an option and they'd like to be able to cast some verbal component spells, then dipping into sorcerer would allow this. However, the downside is needing to forego bard progression in order to do this and this will require a minimum 3 level dip in order to get the Subtle Spell metamagic that would allow bypassing of verbal components.

Simply avoid verbal component spells

This is by far the least attractive option because there are only a few spells under the entire bard list that don't require a verbal component.

This happened at my table, here's what we did and how the players reacted:

In one game I was playing in, this happened to one of the PCs. Our DM ruled that they could not cast a spell with verbal components and they suffered this until we were able to regenerate their tongue.

From a player perspective, the 'maiming' of a PC was awful. It was awful when/why it happened and it was awful for them to have to deal with it for a day until our other cleric could prepare Regenerate. If you are the DM (or anyone reading who is a DM), I would be very wary about mutilating your players and giving them negative mechanical affects. It removes a lot of fun and adds a lot of table issues that can otherwise be avoided.