Yes, the Wyvern is probably intended to have disadvantage.
While Word of God could always overturn this (but it didn't), it seems pretty clear that the intent is for this to apply to reach attacks. They have the key phrase "ranged attack" and use it elsewhere in the rules (e.g. in the Monk's deflect arrows ability). This heavily implies that the more awkward wording is there for a reason.
They might have worded it this way so that ranged attacks made from melee are normal, rather than disadvantaged. But I suspect this case is less likely than the reach attack one (your mileage may vary).
In other words, it's not a case where it looks like they used simple language and forgot about some corner cases, like the interaction between druids and lethal damage.
As to why this would be the case, consider that a large wyvern doesn't have a 10' body with a long head that sits 10' away. The entire wyvern occupies a 10'x10' space, and stretches or pounces when it needs to attack outside of reach. This kind of stretching / pouncing will inherently limit the creature's attack vectors, perhaps in a way that dropping to the ground will help.
In other words, the monster isn't right there. The monster is still 10' away.
Also remember that the monster doesn't have to stay at range. As soon as they drop prone, it can close to within 5' and start taking advantage on its attacks.
From a mechanical standpoint, this makes prone a general strategy to take for things attacking you that you can't reach and/or luring things into melee. This simplifies the mechanical design, and helps avoid loopholes.
As a house rule, you can certainly scale down the effectiveness of the prone condition. It's unlikely to break much with the current corpus of monsters. However, doing so will always compete with simply closing to melee and gaining advantage... And it is unlikely that this will be brought into the official rules (although maybe it will! Sloppy templating, etc.).
Grapples don't involve attack rolls, so the prone and restrained conditions don't give enemies advantage on the ability check
Grapples are a special type of attack, but use an ability check in place of an attack roll. This is clear from the description of grappling in the rules - specifically, the paragraph after the one you quoted:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
[...] Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). [...]
This distinction is reiterated in an official ruling in the Sage Advice Compendium:
When you make a Strength (Athletics) check to grapple or shove someone, are you making an attack roll?
No. That check is an ability check, so game effects tied to attack rolls don’t apply to it. Going back to an earlier question, the hex spell could be used to diminish a grappler’s effectiveness. And if the grappler’s target is under the effect of the Dodge action, that action doesn’t inhibit the grapple, since Dodge doesn’t affect ability checks.
The prone and restrained conditions specifically refer to "attack rolls" being affected. Ability checks, such as Strength (Athletics) checks or Dexterity (Acrobatics) checks, are totally unaffected by these conditions - barring DM fiat.
Rules designer Jeremy Crawford also unofficially confirmed this distinction in a 2015 tweet:
The grapple option in the Player's Handbook is an attack, but it uses an ability check in place of an attack roll.
Does the sage advice on ability checks contradict this? It states that shove & grapple are not attacks?
There's no contradiction. They are unusual attacks that lack attack rolls.
And again in a 2016 tweet that references the official Sage Advice Compendium ruling:
The ruling is correct. Grapple is an odd attack that doesn't use an attack roll.
So no, a prone or restrained creature doesn't cause other creatures to have advantage on an attempt to grapple it.
Best Answer
No, you are not within 5ft and so do not have advantage
The spell states you 'vanish' (whatever that means mechanically), make the attacks, and then after all the attacks are done you can teleport to various spaces. Nothing says you teleport to each target to make the attack. Likewise, you would have disadvantage to hit prone targets.
I agree the flavor of the spell is that you are moving so fast you can run up and hit all the targets, but that is not what the spell says it does. This is similar to how you don't actually use the weapon you are holding to make the attacks. Additionally, don't let the fact that you are making melee spell attacks throw you. You can make the 'melee' attack because the spell says you can, regardless of distance, and even a regular melee attack wouldn't necessarily have advantage vs a prone target (if you were using a reach weapon for instance).