Some of Beasts of Burden are Probably Inherently Evil While Others Are Just Beasts
Firstly, since your question references the movies, I would point out that the movie version of The Hobbit shows a number of different types of animals being used as beasts of burden. Thorin and his nephews ride goats up to Ravenhill. Dain rides a pig. Thranduil rides an Elk. So in movieverse, there are plenty of different types of beasts of burden around.
Next, I think we have to address the creatures that you've referenced one by one.
Trolls are probably evil
In The Two Towers book, Treebeard tells us that Trolls were made by "the Enemy" (likely Morgoth) in mockery of Ents as the Orcs were of Elves.
'Maybe you have heard of Trolls? They are mighty strong. But Trolls
are only counterfeits, made by the Enemy in the Great Darkness, in
mockery of Ents, as orcs were of Elves. We are stronger than Trolls.'
If we assume that "cave trolls" are truly a type of troll, then I think we can assume that they are creatures of the dark forces, a twisted mockery of Ents.
Oliphaunts are probably not evil
The Oliphaunts appear to be associated with the Haradrim men rather than with Sauron or Morgoth. Sam has the following to say about oliphaunts:
But I've heard tales of the big folk down away in the Sunlands. Swertlings we call ‘em in our tales; and they ride on oliphaunts,
‘tis said, when they fight. They put houses and towers on the
oliphaunteses backs and all, and the oliphaunts throw rocks and trees
at one another. So when you said "Men out of the South, all in red and gold," I said "were there any oliphaunts?"
While the Haradrim do fight for Sauron, they are ultimately just men and probably not capable of creating things like orcs and trolls. It is far more likely that oliphaunts are animals that the the Haradrim have tamed. As for why Men of the West don't use them -- it is likely that they aren't an animal that occurs in the wild in the West. They seem associated firmly with the South.
It is unclear whether fell beasts are always evil, but the description of them seems to suggest it.
We know relatively little about the origins of the fell beasts, other than the suggestion they are of "an older world." They certainly seem to be unpleasant beasts from the description we get in Return of the King.
... it was a winged creature: if bird, then greater than all other
birds, and it was naked, and neither quill nor feather did it bear,
and its vast pinions were as webs of hide between horned fingers; and
it stank. A creature of an older world maybe it was, whose kind, lingering in forgotten mountains cold beneath the Moon, outstayed their day, and in hideous eyrie bred this last untimely brood, apt to evil.
Given that ugly, stinky creatures in Tolkien are often evil, this doesn't bode well. The "apt to evil" line is also rather damning, though "apt" to evil isn't quite the same as unambiguously evil. However, there is also the suggestion that the Dark Lord made these creatures even worse by feeding them "fell meats."
And the Dark Lord took it, and nursed it with fell meats, until it
grew beyond the measure of all other things that fly; and he gave it
to his servant to be his steed.
So it sounds like you have creatures who already tended toward evil that Sauron made even worse.
Wargs are probably evil
We don't know much about the origins of Wargs, but Tolkien describes them as "demonic wolves" in a letter to Gene Wolfe.
Dear Mr Wolfe, Thank you very much for your letter. The etymology of
words and names in my story has two sides: (1) their etymology within
the story; and (2) the sources from which I, as an author, derive
them. I expect you mean the latter. Orc I derived from Anglo-Saxon, a
word meaning demon, usually supposed to be derived from the Latin
Orcus -- Hell. But I doubt this, though the matter is too involved to
set out here. Warg is simple. It is an old word for wolf, which also
had the sense of an outlaw or hunted criminal. This is its usual sense
in surviving texts.* I adopted the word, which had a good sound for
the meaning, as a name for this particular brand of demonic wolf in
"Demonic wolves" certainly seems to suggest evil. Also, every time we see the wargs in the books, they are doing evil works. This includes in The Hobbit when they aren't really allied with Sauron, but are still working with the goblins.
As for the beasts that pull Grond ... I have no idea. I believe these are just called "great beasts" and aren't really addressed anywhere.