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.
See? Yes. Notice? Maybe.
The rules on Hiding and stealth give a lot of leeway to a DM, so you are absolutely within your rights to declare that once a creature is looking directly at you, you cease to be hidden from it. But keep in mind a couple of important distinctions:
1. Hiding is different than being unseen
Any creature that is in a heavily obscured area, or invisible, or fighting a blinded enemy is "unseen". But a creature that is hiding is not only unseen, but also unheard, and generally unperceived. They might keep very still, or stand close to similarly colored parts of their surrounding, or compact their body so their silhouette no longer seems humanoid. In short, they are doing more than being out of line of sight.
As evidence of this distinction, consider the following on page 177 of the PHB:
An invisible creature can’t be seen, so it can always try to hide. Signs of its passage might still be noticed, however, and it still has to stay quiet.
To put it another way (and to quote a 2016 Errata in the PHB):
the question isn’t whether a creature can see you when you’re hiding. The question is whether it can see you clearly.
2. You can't Hide from a creature that sees you, but maybe you could remain hidden
The rules are clear that
You can’t hide from a creature that can see you clearly, (PHB, p. 177)
but note that this only rules out becoming hidden: it doesn't necessarily rule out remaining hidden. So as a DM, it will totally be up to you whether or not the Goblin will notice the already hidden rogue when they gain line of sight.
Keep in mind that combat is a chaotic and distracting place. As an example of what that level of chaos can do to perception, note how difficult it is to count how many passes the players in white make in this linked video.
Note, though, that once the hidden rogue attacks anything, they won't be able to hide (from the goblin) again, unless they've moved to some new location that the goblin can't see. The rules are clear that while you could remain hidden while visible, you can't hide.
3. Advantage/Disadvantage (on attacks) is not from being hidden, it is from being unseen
All the points above are meant to indicate that a creature could remain hidden while they are seen: that is, an enemy might not notice them, or might not know where they are. But that does not necessarily mean they will gain advantage on an attack roll against such an enemy.
Any place that the rules suggest a hidden creature will gain advantage on an attack, they justify this as a result of the hidden creature being unseen. Such as:
Master might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack before you are seen. (PHB, p. 177, bold added)
And:
When you take the Hide action, you make a Dexterity (Stealth) check in an attempt to hide, following the rules in chapter 7 for hiding. If you succeed, you gain certain benefits, as described in the “Unseen Attackers and Targets” section later in this chapter. (PHB, p. 192, bold added)
So the only real benefit a hidden character could get against an enemy that can see it is that the enemy might not notice them. This could cause an enemy to be surprised (first round of combat only) or to be unable to intentionally attack the hidden creature (because they don't know it's there). But a creature hidden in plain sight would not get advantage on attack rolls against a creature that could see them, regardless of whether or not they remain "hidden".
Best Answer
So there are two issues to consider, and my original answer only considered one of these. I can't find a definitive answer, but interpreting RAW to answer:
Issue One: Does a Verbal Component Spoil the Hidden Advantage?
Short Answer: GM's Call
There's no general rule I can find which dictates how loud/obvious verbal components are. There's a couple of specific spells, but nothing general. So it's a GM's call as to whether the verbal component can be heard and understood to be an attack. I'd err on the side of the player in most cases, esp if combat is already underway; the noise of combat would probably cover most verbal components.
If the GM rules the noise reveals the warlock, then none of the attacks will have advantage.
Do Separate Attack Rolls from Eldritch Blast All Use Advantage?
Again, there's no RAW which definitively answers this. But we can infer an answer using two other rules:
Attacking while Hidden, PHB, p195
Eldritch Blast Cantrip, PHB, p237
Since you're making a separate attack roll, you give away your location on the first roll, so you'd have advantage on the first, but not on the subsequent rolls. There's no wording to suggest the two (or more) blasts are simultaneous.
As a GM, I can imagine a circumstance where a player might convince me otherwise. But that would probably be outside of RAW.
So, in the specific case of your Goblin Warlock, It's GMs call if the first attack gets Advantage, but the second should definitely not.
A lot of D&D is the GM making calls like these. The rules are pretty good as a baseline, but far from comprehensive.