For example: regarding the question why God created the world, since He needs nothing, being perfect, they answered: because He wanted to bestow good, and it is the nature of the good to do good. But it is still difficult: why did He need to bestow good—what was lacking for Him before He bestowed good by creating the world? Here the answer is, I do not know, because "if I knew Him, I would be Him." By way of analogy: someone arrived on another planet and found a strange machine there. He tried to drive it—it didn't move. He tried this and that, etc. Until he discovered that it could wash clothes. So clearly the engineer and manufacturer built this machine in order to wash clothes; but to the question why the manufacturer made such a machine—was it because his wife asked him to? Or because he wanted to submit the machine as a final project in engineering school?—we would not know, because we do not know the manufacturer. So too here: why did God create the world? "No thought can grasp Him at all." But for what purpose did He create the world—that one can know by reflecting on man and his conscience and on the world and how it operates.

And this is my humble expanded view.

That is a different distinction. When you speak about the Holocaust and ask "for what purpose," you mean what are we supposed to learn from it (as in Maimonides, Laws of Fasts). But what we are supposed to learn from it is exactly what the Holy One, blessed be He, intended to teach us—that is, the reason it happened. For example, the Haredim explain that the Holocaust teaches that one must oppose Zionism, while the Religious Zionists explain that it teaches that one must join Zionism and oppose Haredism. Each side, of course, learns what it thought to begin with (in other words: it teaches nothing). But suppose we accept one of the sides: if the Holocaust came because we opposed Zionism, that means the Holy One, blessed be He, did it in order to teach us that we should be Zionists, and vice versa. So where exactly does the difference between "why" and "for what purpose" come in here? One can of course wonder why it is important to the Holy One, blessed be He, to teach us that we should be Zionists or Haredim, or why it is important to Him to teach us anything at all. But that is a completely different question.
In my personal opinion, He did not cause the Holocaust (Hitler did), and therefore the question does not arise to begin with. He did not teach us anything, nor did He intend to teach us anything. As Rabbi Amital describes his conversation with Abba Kovner: Kovner told him that he lost his faith in the Holy One, blessed be He, in the Holocaust, and Rabbi Amital replied that there he had lost his faith in man. Quite apart from the fact that any sensible person understands that the world follows its natural course, if the Holy One, blessed be He, intended to teach us something, then He failed pedagogically in a resounding way (after all, everyone learns from it what he thought to begin with, aside from a few exceptions like Rabbi Teichtal, etc.). It does not befit an omnipotent being like Him to fail so magnificently. Therefore it is more reasonable that He did not intend to teach us anything at all.
That does not mean one cannot learn things from the Holocaust. One can speak about moral lessons, about the importance of fighting evil, and so on. But not theological lessons.
That is my broad opinion, vast as the sea and deep as puddles.

Much appreciated.

Indeed, I emphasized that we do not know exactly what the Holy One, blessed be He, wants to teach us, and therefore, if I understood Maimonides correctly, the point is that He is hinting to us that we need to correct our ways. What should be corrected? Let each person choose according to his outlook.

And regarding your view that the Holy One, blessed be He, has no involvement here [only Hitler does], how would you explain the words of Maimonides that in every trouble one must repent, and if not, this is cruelty? According to you, there is no divine message in any trouble at all.

The Satmar Rebbe already said: the State is an act of Satan; the Holocaust—from the Holy One, blessed be He? What a wonderful Satan….

The obligation to correct our ways has nothing to do with interpreting calamities. Human psychology is built in such a way that when there are troubles, it is an opportune time for soul-searching and self-correction. Therefore, when there are troubles, the Sages recommend that we examine ourselves. But the obligation to correct our ways does not concern the trouble or its interpretation—neither the "for what purpose" nor the "why." Correct your ways, even if the Holy One, blessed be He, did not cause these troubles at all.
I have no explanation for the words of Maimonides, but there is common sense, which says that the world follows its natural course. And if the Holy One, blessed be He, caused the Holocaust, then Hitler is not guilty (the explanation of double planes is nonsense, just like "effort" and "trust in God." It is absurd babble already on the logical-conceptual level, just like a square triangle).
Therefore, one of two things must be true: either there is a "why" and there is a "for what purpose," and we can also know the answers to those questions; or you do not accept that, in which case the only alternative is to reject the words of Maimonides.
As for me, I suppose that the Holy One, blessed be He, changed policy, and in the past He was more involved. In the past there were also open miracles, and there were prophets as well, whereas today there are neither. In other words, there is a clear policy change according to all views, and therefore in my opinion there is no reason not to say that there was also a policy change regarding His involvement in the world in general (hidden miracles). If you want to see my views on this in more detail, you can see them here:

חיפוש אחר אלוהים בעולם

As for providence in general, I discussed it at length, to my misfortune, in the second book of my trilogy, after which I was officially declared a first-rate heretic.

Those rabbis think that they themselves are righteous, and that the troubles came because of others.
They cannot free themselves from the outlook that justifies everything they think, and they are unable to consider that perhaps their entire approach is bad and contrary to the Torah.

And why are they unable to think or say that they are mistaken?
Mainly because it hurts their livelihood. And also because a person does not declare himself wicked.