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.
They communicate with you directly
Let's clear up the underlying assumption. The flock of familiars spell is still cast by you. The relevant part of the find familiar's description says (emphasis mine):
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.
Meaning the spell is still cast by you and so the additional familiars are yours. (This also means it only summons two of them [assuming cast at 2nd level] since you have a familiar already.)
So, because they are your familiars they can communicate telepathically with you (the Warlock), but they don't have to (and gain no ability to) do it via the preexisting familiar.
As they are summoned and communicate by you, their telepathy-range (of 1 mile) is counted from you, not your familiar even if you cast flock of familiars through your familiar.
In terms of senses they use the same rules as find familiar which includes using your action to perceive though the senses of one familiar, as long as that familiar is within 100 feet of you.
As more a side issue, there is some ambiguity as to what form the new (temporary) familiars can be. The flock spells says it can be any form of a find familiar familiar and Pact of the Chain reads:
When you cast [find familiar], you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.
--- Player's Handbook, p. 107
As Pact of the Chain specifically refers to casting the spell and not just to "familiars you summon" it could be argued the new forms aren't available to the flock of familiars spell. I personally would rule otherwise1 and you should check with your DM which way they're going to rule.
An additional possible concern, you can cast flock of familiars through you familiar, but is unclear what effect it will have. Specifically, it has a range of touch, but does not refer to the target of the spell at any point. It also doesn't clarify where the familiars appear (such as "an unoccupied space within range"), so it is possible the intent is your familiars appear "from your touch", but it is not clear. If we assume (or rule) this, casting it though your familiar would let them appear at it, but have no effect beyond that.
1: On the simple basis that it is cooler that way (Rules as Fun) and that Pact of Chain doesn't account for multiple ways to gain familiars, because at time of printing there weren't any other than find familiar (Rules as Intended).
Best Answer
It depends on the item.
Rod of the Pact Keeper only increases the DC for your warlock spells:
Rod of the Pact Keeper only raises your spell save DC when you are casting one of your warlock spells. So Rod of the Pact Keeper would not increase the DC for your familiar, unless you are casting one of your warlock spells through the familiar while using the Rod of the Pact Keeper.
For an item to work with this, it would have to read something like:
without specifying a use case, such as the Robe of the Archmagi:
Unlike Rod of the Pact Keeper, Robe of the Archmagi does not specify when or for what your spell save DC is increased. It just increases. For everything.
For an example of another feature where Robe of the Archmagi works and Rod of the Pact Keeper does not, let us examine the Staff of Charming:
In particular, command is not a Warlock spell. So when casting command from the Staff of Charming, we would not benefit from the Rod of the Pact Keeper's bonus, since command is not one of our warlock spells, but we would get the bonus to spell save DC provided by Robe of the Archmagi.