If Sauron was Morgoth's most trusted lieutenant, why did everyone take his rings?

Question

When Morgoth was defeated, Sauron - Morgoth's most trusted lieutenant - created the rings of power. At least according to the One Wiki to Rule Them All.

Why did everyone welcome these gifts from someone known to be an ally of Morgoth?

Answer 1

Because they didn't know he was Sauron.

From Of the Rings of Power and the Third Age:

Of old there was Sauron the Maia, whom the Sindar in Beleriand named Gorthaur. In the beginning of Arda Melkor seduced him to his allegiance, and he became the greatest and most trusted of the servants of the Enemy, and the most perilous, for he could assume many forms, and for long if he willed he could still appear noble and beautiful, so as to deceive all but the most wary.

He was quite successful in this deception, and only Elrond and Gil-galad were not taken in by him.

... long he sought to persuade the Elves to his service, for he knew that the Firstborn had the greater power; and he went far and wide among them, and his hue was still that of one both fair and wise. Only to Lindon he did not come, for Gil-galad and Elrond doubted him and his fair-seeming, and though they knew not who in truth he was they would not admit him to that land. But elsewhere the Elves received him gladly, and few among them hearkened to the messengers from Lindon bidding them beware; for Sauron took to himself the name of Annatar, the Lord of Gifts, and they had at first much profit from his friendship.

Elsewhere (in Unfinished Tales), and as the character of Galadriel developed, it's told that she also was not deceived and was in fact his chief opponent:

In Eregion Sauron posed as an emissary of the Valar, sent by them to Middle-earth ("thus anticipating the Istari") or ordered by them to remain there to give aid to the Elves. He perceived at once that Galadriel would be his chief adversary and obstacle, and he endeavoured therefore to placate her, bearing her scorn with outward patience and courtesy.

The reason is given in a footnote to the text as follows:

The note goes on to say that Galadriel was not deceived, saying that this Aulendil was not in the train of Aulë in Valinor, "but this is not decisive, since Aulë existed before the 'Building of Arda,' and the probability is that Sauron was in fact one of the Aulëan Maiar, corrupted 'before Arda began' by Melkor."

Despite all of this he was very successful in deceiving the other Noldor:

Sauron used all his arts upon Celebrimbor and his fellow-smiths, who had formed a society or brotherhood, very powerful in Eregion, the Gwaith-i-Mírdain; but he worked in secret, unknown to Galadriel and Celeborn. Before long Sauron had the Gwaith-i-Mírdain under his influence, for at first they had great profit from his instruction in secret matters of their craft.

What's particularly ironic about this is that it's exactly the same weakness in the Noldor that Melkor had exploited when he deceived them in Valinor, as The Silmarillion tells:

But the Noldor took delight in the hidden knowledge that he could reveal to them; and some hearkened to words that it would have been better for them never to have heard.

So he didn't openly announce himself as Sauron but claimed to be someone else, in some versions of the stories an ambassador from the Valar, and it was only after he had forged the One Ring that the Elves knew they had been deceived; again from Rings of Power:

As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and perceived that he would be master of them, and of all that they wrought.

There is one incorrect statement in your question - that Sauron had created the Rings of Power. In actual fact he didn't; the only Ring that Sauron created was the One. The Elves created the other Rings but with the aid of his instruction; again from Rings of Power:

In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power. But Sauron guided their labours, and he was aware of all that they did...

Now the Elves made many rings; but secretly Sauron made One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last.

As for the other races (Men and Dwarves), they accepted their Rings because (1) Men were more easily dominated by Sauron, and (2) both Men and Dwarves found the Rings useful. Rings of Power is once more our source:

Seven Rings he gave to the Dwarves; but to Men he gave nine, for Men proved in this matter as in others the readiest to his will.

The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows. They used their rings only for the getting of wealth; but wrath and an over-mastering greed of gold were kindled in their hearts, of which evil enough after came to the profit of Sauron.

Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth...

Although it's not stated explicitly, it seems reasonable to assume that Men and Dwarves would have also had somewhat less knowledge of exactly who they were dealing with too.

Answer 2

When he offered to help make the Rings it was under the alias of Annatar, Lord of Gifts. A few Elves were suspicious (Gil-galad and Galadriel), but Celebrimbor and the smiths of Eregion chose to go along with the offer. Even the suspicious elves did not truly understand who he was until he made and put on the One Ring though.

For Men he more likely gave them to people were less likely to even care; Black Númenóreans, or men of the east and south.