I think that the answer to this question is a cautious and nuanced "yes". Consistently through the history of Arda, the Valar do a pretty poor job of administration:
Originally,
But when the Valar entered into Eä [Arda] they were at first astounded and at a loss, for it was as if naught was yet made which they had seen in vision, and all was but on point to begin and yet unshaped, and it was dark. For the Great Music had been but the growth and flowering of thought in the Timeless Halls, and the Vision only a foreshowing; but now they had entered in at the beginning of Time, and the Valar perceived that the World had been but foreshadowed and foresung, and they must achieve it. So began their great labours in wastes unmeasured and unexplored, and in ages uncounted and forgotten, until in the Deeps of Time and in the midst of the vast halls of Eä there came to be that hour and that place where was made the habitation of the Children of Iluvatar.
To me, this sounds a bit like a poorly thought out "Wow! That looks like fun! Let's go!" and when they get there, it's "Huh? Where's all the fun stuff?"
So, they build Arda, yet are unable to prevent Melkor [Morgoth] from adding a strain of corruption into the very fabric of Arda (see HoME: Morgoth's Ring for much more on this) which poisoned the world for all time.
And in this work [of building Arda] the chief part was taken by Manwe and Aule and Ulmo; but Melkor too was there from the first, and he meddled in all that was done, turning it if he might to his own desires and purposes; and he kindled great fires.
and
Yet it is told among the Eldar that the Valar endeavoured ever, in despite of Melkor, to rule the Earth and to prepare it for the coming of the Firstborn; and they built lands and Melkor destroyed them; valleys they delved and Melkor raised them up; mountains they carved and Melkor threw them down; seas they hollowed and Melkor spilled them; and naught might have peace or come to lasting growth, for as surely as the Valar began a labour so would Melkor undo it or corrupt it.
They keep working (without dealing with Melkor) until Tulkas joins the Valar.
...the First War began before Arda was full-shaped, and ere yet there was any thing that grew or walked upon earth; and for long Melkor had the upper hand. But in the midst of the war a spirit of great strength and hardihood came to the aid of the Valar, hearing in the far heaven that there was battle in the Little Kingdom; and Arda was filled with the sound of his laughter. So came Tulkas the Strong, whose anger passes like a mighty wind, scattering cloud and darkness before it; and Melkor fled before his wrath and his laughter, and forsook Arda, and there was peace for a long age. And Tulkas remained and became one of the Valar of the Kingdom of Arda; but Melkor brooded in the outer darkness, and his hate was given to Tulkas for ever after.
They build the two Lights, but Morgoth destroyed them. When? While the Valar are partying!
Now it came to pass that ... Manwe ordained a great feast; ... But Aule and Tulkas were weary; for the craft of Aule and the strength of Tulkas had been at the service of all without ceasing in the days of their labour. And Melkor knew of all that was done, for even then he had secret friends and spies among the Maiar whom he had converted to his cause; ... Now therefore the Valar were gathered upon Almaren, fearing no evil, and because of the light of Illuin they did not perceive the shadow in the north that was cast from afar by Melkor; for he was grown dark as the Night of the Void. And it is sung that in that feast of the Spring of Arda Tulkas espoused Nessa the sister of Orome, and she danced before the Valar upon the green grass of Almaren.
Melkor destroys the Lights and what do the Valar do?
the Valar fortified their dwelling, and upon the shores of the sea they raised the Pelri, the Mountains of Aman, highest upon Earth. ... upon whose summit Manwe set his throne.
So, the Valar are powerful enough to defeat Melkor: Manwe alone is nearly his equal, and Melkor flees from Tulkus. And what do they do? They let Melkor have uncontested control of Middle-earth while they live in their paradise of Aman.
And why don't they chase him and subdue him? Because they did not know where or when the Children would first arise and feared that their making war upon Melkor might disturb the Children. So...they leave Melkor in complete control of Middle-earth where the Children would arise. Good strategic planning, Manwe!
Strategy: Hold off on the destructive war that must come until you're sure that the Children of Iluvatar will be around to be caught up in it!
The Children come and the war comes and it's a mess, but (some) of the Children come to Aman and Melkor is chained and everyone lives happily ever after.
But after those three ages, Manwe judges Melkor and sets him free within Aman. (Note that Manwe makes this hugely consequential decision without asking Mandos -- who knows roughly what's coming -- for advice. Who's to blame there? Both of them! Manwe for not asking and Mandos for not tugging on Manwe's sleeve and saying, "Umm, Boss. There's something you need to hear. For Eru's sake, ask me about this!"
Melkor goes free, stirs up unrest among the Noldor, leaves Aman unbidden, locates Ungoliant (Why did the Valar leave her alone? Why did they not look into the dark nooks and crannies of Aman?), returns, steals the Silmarils, poisons the Two Trees, plunges Aman into darkness, and flees back to Middle-earth.
What were the Valar doing while their great enemy is on the loose again? Party time!
Now it was a time of festival, as Melkor knew well ... it was then the delight of the Valar ... to clothe themselves as in a vesture in the forms of the Children of Iluvatar, so also did they eat and drink, and gather the fruits of Yavanna from the Earth, which under Eru they had made.
Feanor excites the Noldor to head back to Middle-earth to retake the Silmarils and commits great evils along the way. What do the Valar do? Curse the Noldor! Raise up the Mountains of Aman even higher. Outside them, build a barrier of enchanted isles to trap mariners travelling West to Aman!
...leaving Morgoth (and he was now known) in control of Middle-earth.
(It's worth noting that Morgoth finds many powerful supporters -- Sauron, Balrogs, Dragons when he get ack to Middle-earth because the Valar three ages previously didn't bother to do a proper clean-up after they defeated him the first time.)
The Valar told the Eldar that they did this because their previous fight with Morgoth was so destructive. Apparently, they preferred that all the peoples of Middle-earth be forever subject to Morgoth (who would not doubt treat them well) than to risk another war.
(A war which comes, anyway five hundred years later when Earendil sneaks through the Valar's barriers and begs for help.)
Is there anyone who could not have seen that the Valar picked nearly the worst possible option?
Jump ahead an Age and consider Numenor. The Valar -- immortal, super-human, guardians of Ea -- seem to let thing get worse and worse on Numenor and leaves it to the Elves to try to turn things around. Finally,
But the fleets of Ar-Pharazon came up out of the deeps of the sea and... at last Ar-Pharazon came even to Aman, the Blessed Realm, and the coasts of Valinor; and still all was silent, and doom hung by a thread. For Ar-Pharazon wavered at the end, and almost he turned back. His heart misgave him when he looked upon the soundless shores and saw Taniquetil shining, whiter than snow, colder than death, silent, immutable, terrible as the shadow of the light of Iluvatar. But pride was now his master, and at last he left his ship and strode upon the shore, claiming the land for his own, if none should do battle for it. And a host of the Numenoreans encamped in might about Tuna, whence all the Eldar had fled.
Then Manwe upon the Mountain called upon Iluvatar, and for that time the Valar laid down their government of Arda.
As well they should, since they were doing a pretty poor job of it!
Finally to the question at hand, what about the Ring?
(1) The Valar can be counted on to make poor choices, usually taking the easy way out until absolutely forced to act.
(2) The Ring was the work of a single Maiar (Maiar are similar to, but of lower power than the Valar). What's more, the work of a Maiar who had been corrupted by Morgoth and had wasted some chunk of his original being -- corruption, doing evil, always diminishes the essential being of the person doing it.
(3) As far as Sauron goes, he's a (comparative) pipsqueak, who was defeated by (a) Luthien and Beren, (b) Numenor, and (c) The Last Alliance. Sauron can plainly be defeated without the destruction of Middle-earth.
I do give the Valar credit for sending Gandalf, though interview process for the other Istari seems to have been flawed.
But other than that, it would have been trivial for them to take the Ring -- send Ulmo to the Brandywine and he carries the Ring back to Aman -- and even if Sauron's former master Aule can't unmake the Ring, Sauron is now just a powerful wielder of merely physical force and even a small army from Aman combined with the forces of Middle-earth would have been enough to defeat him and haul him off to the Halls of Mandos in Aman.
Perhaps this time they would be a bit more careful with their prisoner?
Bottom line: The Valar's record in the management of the evils of Middle-earth is wretched. Their decision to refuse the Ring is just the last documented case of a long series of misjudgments on their part.
