Because it was Sergei Korolev's dream.
Korolev was the Soviet's "von Braun" except, of course, he was Russian, not German.
Here is an excellent docudrama about von Braun and Korolev. Anyone who is truly interested in the history of the Soviet/American space race should set aside 90 minutes to watch because it is well worth the time spent....
https://www.youtube.com/watch?v=zWumRGO1420
Thank you for digging this up. I saw this video. I couldn't find it when I posted. I also forgot Korolev's name.
I don't think that either Braun or Korelev would support for an Apollo fraud. In fact, most of the Nazi scientists would support such a fraud. I don't think their was any fraud at all. If there is one thing one could say about the Nazis, then it is that they were honest. Tactical deceits, yes. Strategic falsehoods, never!
I was wrong when I used the word 'conspiracy' before. I meant more like 'quid pro quo'. I don't think there was any direct communication between Korelev and Warner. Having common backgrounds, they could deduce what they other was thinking from the others actions. Certainly there was no oral or written agreement. I speculate there was what lawyers would call 'quid pro quo'. Y
Also, I know that the 'military industrial complex' (MIC) is not monolithic. I fully agree that most of the people managing and owning the corporations had families, friends and fairly benign ideologies. The majority of what we call the MIC would be deeply devastated by a nuclear war. However, a nuclear exchange would not affect everyone the same degree.
No one had shown, or has shown, that a nuclear war would really wipe out the human species. There were lots of movies showing it. That was a speculation. Everyone in Europe and Asia knew that a society could be rebuilt even if 80% of its people were wiped out. I got that percentage from 'Mein Kampf', by the way.
Hitler said that to prevent overpopulation, 80% of all human beings in each generation should die violently one way or another while the rest reproduce like jack rabbits. The survivors would be the master race. They needn't even be Germans! Every member of the SS, from Braun to Korelev, were required to read Mein Kampf . They were expected to take it seriously. I speculate that both Braun and Korelev wanted their families to be part of the remaining 20%.
I further speculate that a lot of Nazi scientists wanted their families to be part of the 20%. The master race must survive. That is why so many of had such large families

Braun had a lot of children!
Lets do a cost-benefit analysis of what nuclear war would mean. I am not just talking about money. I acknowledge that the main cost would be in people, not money. Ask your selves what part of the MIC would NOT be devastated if nuclear war had really broken out between the U.S. and Russia. I maintain that it was those people living near the White Sands Missile Range that would be least threatened by an actual nuclear war.
Many of the German scientists lived in the region near the White Sands Missile Range. If a nuclear bomb hit New York or Washington, DC, no of Brauns family would be killed. El Paso and Las Cruces could be targets. What, Las Cruces a target?

However, these are not high density populations. A nuke hitting El Paso would not kill as many people as a nuke that hits New York. Most of the scientists probably lived in the suburbs anyway.
I speculate a similar situation existed in Russia. It is unlikely that most of Korelev's family lived in Moscow. If Braun managed to get the U.S. to nuke Moscow, like he speculated in his novel, it is unlikely that any Korelev's would be killed. Moscow is occupied mainly by Slavs, who Mein Kampf tells us are subhuman. So relatively few families of the Master Race would be killed by a one nuke on Moscow.
So Korelev and and Braun are doing a cost-benefit analysis of a heated cold war. Their main criteria is the probable cost and benefit to their immediate families. Maybe they have a slight concern about the other ones family, since they probably knew each other. They do not know many people in the America or Russia.
The Nazis don't even know many German Americans. Braun has no incentive to protect the German communities already set up in the United States. None of them are HIS relatives. His German community is concentrated in a area full of Hispanics, Jewish refuges, and a few ranchers who will be pushed out anyway. In the mind of some Nazis, the German American is a traitor.
Let us consider the cost an benefits of an arms race.
Suppose there was no arms race at all. Braun, Korelev and community have no jobs or social status. Their families are poor. They are also rejected by their societies. Their may be no community since there is no focus to keep the different German families together. Definitely one doesn't want that.
Suppose their is an arms race with a nuclear war. Lets us imagine a really bad one, where 80% of the population in each community is wiped out. Most of it would be in the urban centers. The Nazi families would mostly be untouched in farm communities with lots of jobs. Hey, no labor. Not very pleasant prospect, but it has a future.
Suppose their is an arms race with NO nuclear war. Lots of jobs for the families. Lots of money. Lots of social status. They became American heroes. Some of them even buy land. Room for their hamilies to grow. And grow and grow.
Maybe some of them could ind jobs on Mars, if a colony was set up? I know that Braun thought space colonization would benefit mankind. He may have even thought of it as a non genocidal way to alleviate the population problem. So if the arms race lead to space colonization, maybe there would be no need for genocide!
So the focal point of the arms race for him would be the colonization of space. The nukes, if necessary, would alleviate the short term problem of over population.
Now, these things a very complex. I am not going to say that it was all planned by Braun and Korelev. They were only two people who couldn't make people do what they wanted. Obviously, there were other factors that greatly amplified their efforts. If people didn't already feel unsafe for other reasons, then the Nazis couldn't get them to go along so easily.
I only point out that as far as Braun and Korelev are concerned, an Apollo moon fraud would be counter productive. It is exactly the wrong thing for their families. A fraud gives them nothing to do and achieves no long term goals. Family dynasties don't really thrive on frauds.
The fraud would have to start from the very beginning to be effective. One would have to make a story for each penny spent on developing manned flight. So I don't think a Apollo fraud would be supported by Braun or Korelev. There is no benefit to their families and lots of potential cost.