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.
It's almost correct, but not quite.
There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.
The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).
The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.
Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.
It's not any more powerful than other options and should not be limited.
Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.
So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.
Best Answer
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):
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:
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).