They were probably inhabited by evil spirits
I want to acknowledge that this answer draws liberally from the answer by Rand al'Thor to one of the linked questions, and the answer by user8719 to another linked question.
Wargs, wolves and werewolves
The August 1967 draft of a reply to "Mr. Rang" includes a note on the origin of the word "warg" that identifies them as "an evil breed of (demonic) wolves"
The word Warg used in The Hobbit and the L.R. for an evil breed of (demonic) wolves is not supposed to be A-S specifically, and is given prim. Germanic form as representing the noun common to the Northmen of these creatures.
The Letters of JRR Tolkien, Letter 297
In The Hobbit, Tolkien tells us that "Wargs" is a name of "evil wolves". As the book is aimed at children, this may be a simplified explanation.
But even the wild Wargs (for so the evil wolves over the Edge of the Wild were named) cannot climb trees.
The Hobbit Chapter 6: Out of the Frying-pan into the Fire
In The Lord of the Rings, Gandalf implies that wargs and werewolves are different creatures.
‘Because these horses are born and bred to the service of the Dark Lord in Mordor. Not all his servants and chattels are wraiths! There are orcs and trolls, there are wargs and werewolves; and there have been and still are many Men, warriors and kings, that walk alive under the Sun, and yet are under his sway. And their number is growing daily.’
The Lord of the Rings Book 2, Chapter 1: Many Meetings
The term "werewolf" does not appear in The Hobbit or The Lord of the Rings, (except the quote shown above) but it does appear in The Silmarillion where they are associated with Sauron and their Sindarin name is given as "gaurhoth":
Sauron was become now a sorcerer of dreadful power, master of shadows and of phantoms, foul in wisdom, cruel in strength, misshaping what he touched, twisting what he ruled, lord of werewolves; his dominion was torment. He took Minas Tirith by assault, for a dark cloud of fear fell upon those that defended it; and Orodreth was driven out, and fled to Nargothrond. Then Sauron made it into a watchtower for Morgoth, a stronghold of evil, and a menace; and the fair isle of Tol Sirion became
accursed, and it was called Tol-in-Gaurhoth, the Isle of Werewolves.
The Silmarillion: Quenta Silmarillion Chapter 18: Of the Ruin of Beleriand and the Fall of Fingolfin
Later in The Silmarillion, werewolves are described as being "fell beasts inhabited by dreadful spirits" that Sauron "had imprisoned in their bodies"
Therefore an army was sent against him under the command of Sauron; and Sauron brought werewolves, fell beasts inhabited by dreadful spirits that he had imprisoned in their bodies.
The Silmarillion: Quenta Silmarillion Chapter 19: Of Beren and Lúthien
Tolkien (as narrator) identifies the creatures that attack the Fellowship in Hollin as "Wargs"
A great host of Wargs had gathered silently and was now attacking them from every side at once.
The Lord of the Rings Book 2, Chapter 4: A Journey in the Dark
Gandalf appears to identify them as "gaurhoth". First, he refers to their captain as "Hound of Sauron"
Gandalf stood up and strode forward, holding his staff aloft. ‘Listen, Hound of Sauron!’ he cried.
The Lord of the Rings Book 2, Chapter 4: A Journey in the Dark
Secondly, the spell he uses against them mentions "gaurhoth"
Naur an edraith ammen! Naur dan i ngaurhoth!
The Lord of the Rings Book 2, Chapter 4: A Journey in the Dark
How do these terms relate to each other
- I see no reason to assume that "wolf" refers to anything other than the ordinary wolf that is found in the wild today.
- The "gaurhoth" or "werewolves" of the First Age presumably are ordinary wolves that Sauron caused to be inhabited by "dreadful spirits".
- The "Wargs" that attack the Fellowship are identified by Gandalf as "gaurhoth". However, Gandalf has also implied that wargs and werewolves are different. This suggests that wargs and gaurhoth are different but related in some way.
So why did the bodies of the Wargs disappear
I don't think that simply identifying Wargs as werewolves (as the term is used in popular culture) explains the dissapearance. Disappearing is not one of the features of a werewolf.
I think a reasonable in-universe explanation is that Third Age Wargs are descended from First Age gaurhoth. Out of universe, it is possible that Tolkien changed his mind about the names. In that case, Wargs are probably also inhabited by dreadful spirits and that is what allows them to disappear.