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.).
Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:
Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.
"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.
This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.
Best Answer
Inspiration gives you advantage(PHB, 125):
But, as you know, if you have both advantage and disadvantage, you count as having neither(PHB, 173):
There is no special exception for advantage that comes from inspiration. It uses the same wording as many other features that grant advantage, without providing any extra rules.