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Given that Sauron spent a whopping 47 years (Second Age 3262-3319) in Númenor corrupting the people and making them do unspeakable things (burning Nimloth, human sacrifice, Morgoth worship, slaughtering the wild men of Middle-earth, the Great Armament), why didn't the good Powers stop him at the outset?

One wonders if the Powers truly cared about the race of Men, the Secondborn Children of Eru...

There are numerous reasons I elaborated below. And there is also one meta: Why any storytelling works do not eliminate the evil immediately at the beginning? That would set the story unrealistic (compared to human experience) and ultimately there would be no story at all. (Your question could be even rewritten into deeper one: Why Eru didn't destroy Melkor yet before he had chance to disrupt anything in Eru's world?)

Details:

What you describe reflects the fact that intervening by "good powers" is never a straightforward matter. Even in daily reality, if after each transgression of laws there came immediate law enforcement response (from "Eru", from police, from other authorities, whomever...), everyone would end as a kind of a criminal after some time. How do we set a clear line for distinction which, as you wrote, "unspeakable things" are still tolerable and which are not? This always leads to a complex discussion with no end. Also, if the "corrections" would came too quickly, there would be no space for character development through challenges (which is not only a matter of plot construction, but an important aspect of real life, what in turn makes the story more believable).

From supposed Eru's perspective, there is no need to interfere if the things in bigger picture are still under Eru's control and cases of the corruption can be still contained without destroying the entire civilization. The mightier the Powers are, the more space for tolerance they give. Actually, from some viewpoint, there were interferences of Eru inside the books, in those uncountable "details" where protagonists were just "very lucky" to effectively hide at the last moment or to hear something important from the enemy at the right time, to inadvertently find something valuable, to survive uneven battles, to meet key person in the middle of the barren lands/woods etc. So from this perspective, we cannot deny that what happened was influenced. There is even a pun to it in the last chapter of Hobbit (see below). And today, there is a famous quote of Albert Einstein: Coincidence is God's way of remaining anonymous.

“Then the prophecies of the old songs have turned out to be true, after a fashion!” said Bilbo.

“Of course!” said Gandalf. “And why should not they prove true? Surely you don’t disbelieve the prophecies, because you had a hand in bringing them about yourself? You don’t really suppose, do you, that all your adventures and escapes were managed by mere luck, just for your sole benefit? You are a very fine person, Mr. Baggins, and I am very fond of you; but you are only quite a little fellow in a wide world after all!”

“Thank goodness!” said Bilbo laughing, and handed him the tobacco-jar.

The first thing you need to understand about Tolkien's writing is that even though he was being influenced by Norse mythology, he himself was a Catholic and that shaped what he wrote. It's why he waffled so much on the nature of orcs, because Catholic theology said that the Devil couldn't create but he needed the orcs to be a race of disposable mooks with no moral agency for the plot. It's why Gandalf got a power upgrade after his death and resurrection, because contact with holiness topped up his spiritual tank.

And in Catholic theology, God is not a micro-manager. He allows humankind to choose our way into corruption. Eru (who is God) would not directly intervene or allow his servants to intervene until they were being directly challenged.

For Eru, the history unfolds as it must since the music has been played and history will follow it.

In the published books, Eru is a bit of a jerk here: Melkor's dissonances made Eru's themes stand out better in the music; the consequence is that Melkor's evil will make the story of the world better because it makes Eru's work stand out better.

There's unpublished work with a prophesy that there will be a second Music eventually, and everything that was wronged will be made right. If that's what Tolkien intended (which is unclear), then Eru is accepting a merely temporary evil to make a better story.

For the Valar, the answer is easy: They cared, a lot actually, but Eru had told them to not interfere with mortals.

They were also generally reluctant to punish evildoers, for a multitude of reasons: Partly because they had not yet seen how bad the evildoers really were, partly because they didn't want to overdo punishment and be evil themselves, partly because they knew that any direct action would come with a lot of collateral damage, partly because the amount of evil throughout history was predetermined by the amount of dissonance in the music anyway (the last reason is my speculation, the others have been alluded to in various places in the Silmarillion).

BTW none of these reasons have any relation with Free Will. I don't know if Tolkien ever cared about that specific debate, and as far as I'm aware of his stories, the issue never arises.

The protagonists in his stories generally don't theorize much about ethics; they fail to achieve their goals mostly due to hubris, with a good dose of mistrust, jealousy, and falling to the deception of the evildoers.