One of Tolkien's own speculations about them is that they fell subject to
Sauron and were eventually destroyed by them. So it's really not that
far-fetched an idea. Ultimately, both might have happened. They
began their missions, strayed into liking the reverence they got from the
Easterners, set themselves up as gods of their own cults, were noticed, then
flattered and seduced by Sauron, became his "allies," and then were destroyed
by him, because allies were only useful to him until they had fulfilled their
purpose in his own plans. In this case, he may simply have disliked them
doing anything that appeared to be setting up competition to him (in this case
in the matter of worship), and his purpose in making them his friends was to
get them close enough to get rid of them.
IMHO, of
course.
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We need the faith to go a path untrod,
The power to be alone and vote with God.
--Edwin Markham