Yes, you can indeed cast the spell. The prerequisites are met by satisfying all of the conditions.
First off, the prerequisite of your familiar being able to cast a spell is covered adequately by Find Familiar. The familiar uses it's reaction, and you use your attack roll. This means YOU are casting the spell while the familiar simply physically delivers it, hence the touch requirement. The important part of this is that it establishes that your familiar CAN cast your spells as an extension of you.
Second, it's your voice, not your familiars. The familiar is not mimicing you, you are speaking and using it as a conduit.
Third, so long as your familiar is within range of the spell for the target, you are within range to cast using the familiar. Since you are casting it from the point originating from your familiar, this is clearly the origin of the spell. Yes your character is the one casting it, however you are looking, speaking and performing your actions through your familiar. This is very similar to how domination works.
Fourth, the flavour text does matter. “The particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion.” Only something capable of touching the weave could cast the spell. Otherwise spells without a verbal component could be cast by anybody as well. So if the requisite is being able to speak the spell into action because you as a caster are capable of touching the weave, then yes, with his pact he can do it through his familiar.
Fifth, all of the conditions to be able to cast through his familiar using a verbal spell are met, bar none. There's nothing preventing the player from being creative and doing this. In fact, if you consider utilizing gates and casting spells through portals, you need to understand that it is in effect the exact same thing as casting through the eyes of a familiar.
Bottom line: His class lets him utilize his familiar as an extension of himself. This pact trumps Find Familiar in that it adds on another specific ability which grants him access to using Verbal spells through his familiar as well, because his familiar is a conduit for him to utilize his voice, which Find Familiar does not normally provide.
This is a very creative use of his pact, and certainly isn't game breaking. While I see numerous No answers on this page, none of them have actually provided substantiation for why the answer is no. If the familiar can cast a touch spell, then the addition of the voice with a pact would obviously grant it the ability to cast a verbal only spell. The conditions are clearly met.
The touch portion of Find Familiar is there to demonstrate that familiar can cast spells as if they were you. Since you are already in contact with your familiar telepathically thanks to Voice of the Chain Master at all times (that you should desire) so long as you are on the same plane. This is what actually makes a POC warlock an interesting choice, and is a completely different playstyle from a typical caster. People just seem hung on FF, but this pact modifies it to function in a different manner, and is specific with reference to the voice.
Since FF is cast as a ritual, you don't need to burn a spell getting your familiar out. Think along the lines of hunter pets. The addition of the Voice is what makes the Verbal only spells a viable option now. It's your voice, says it right in the description. Since it's your voice, that makes it your spell. FF already established that a familiar can cast your spells, and there's nothing preventing you from casting from your familiar anywhere in the rules. The thing here is that people are still treating it like FF, but ignoring the Voice of the Chain Master, which is the important part that the OP is actually asking about.