Unfortunately, this is not the first time that askhistorians has had difficulty determining the origins of a fascinating and possibly otherwise unreported statement found on the USHMM website. A few years ago, u/commiespaceinvader and I worked to determine the origins of English language reports that homosexual camp survivors were transferred to prisons to serve the remainder of their sentences (if they did indeed have a remainder to the sentence). The first attempt is found in a secondary question on the thread here (I failed), but the communist invader from space had my back here. In this post, CSI addressed why the study of the Nazi assault on homosexuals has been difficult to accomplish.

Of course, none of this has anything to do with your question, except that the USHMM policy of not citing little known information in its encyclopedia resource entries has now led to such difficulties twice. (EDIT: SEE my other comment that contains the response of the USHMM to my inquiry. Lesson? Contact the USHMM, they will write back.) If you notice in the first link, when I did an online search, I could only find two sources for the statement that homosexual men were reincarcerated by the Allies. One was the uncited USHMM statement and a number of other sites which simply copied them and the other was another site which cited a broken website of from its own organization. A similar difficulty was found in this case.

In my, admittedly, cursory search of the internet to determine the origins of such a US policy, I have been unable to find its mention. Only websites copying the USHMM discuss it. All that to say, I cannot answer at this moment what US policy was or exactly why it was policy. Thus, I will try to give context and suggest the likely reason.

Liberation and the Jews

First, the phenomenon on which you stumbled is not an unusual one. Death Mills is not the only report on the liberation of camps which ignores or downplays Jewish victimization. Notably, Edward Murrow’s famous broadcast from Buchenwald on April 16th, 1945. In it, Murrow never mentions Jews in his report. He notes a German communist, Czechoslovakians, and children, who Murrow notes were seen by the Germans as “enemies of the state.” When suggesting the nature of the men interned in the camp, he simply stated “Men kept coming up to speak to me and to touch me. Professors from Poland, doctors from Vienna, men from all Europe. Men from the countries that made America.” This formulation was not accidental, but a trend in American representations of the Nazi atrocities.

In Death Mills, the United States attempted to create a film that would demonstrate, first to Germans as Die Todesmühlen and then to Americans, the nature of Nazi atrocities. The footage used came from US signal corps film and that of their allies. In spite of knowing of both the concentration camps and even the extermination camps as early as 1942, Americans did not understand the nature of the German camps and policies. Films such as Death Mills and the photographs and stories published in such sources as Time Magazine shocked Americans IN SPITE of the earlier reports. Pictures truly were worth a thousand words.

Just as with Murrow’s broadcast, the film never specified Jews as victims. Instead, the film stated that “ Those who survived could answer the roll call of all the nations of Europe, of all religious faiths, of all political beliefs; condemned by Hitler because they were anti-Nazis.”

The question then becomes why these reporters did not discuss the atrocities against the Jews.

As an important aside, the USHMM page that you note states, "Nonetheless, in compliance with U.S. policy, the word 'Jew' is never used.” It is possible that the USHMM statement could mean general policy rather than specific policy. The use of the term “compliance” makes this less unlikely, but I do think it points to a difference that we may see in this answer. A government does not have to make a specific, written rule for a policy to guide those who execute its wishes. Thus, whether or not there was a policy document specifying the avoidance of listing Jews as victims, there was cultural and political pressure to do so. A kind of unstated policy. More on this later.

It may be obvious, but the Holocaust—here meaning the murder of Europe’s Jews—was complicated. The Nazis and their agents accomplished it a variety of ways, at a variety of times, in a variety of locations. It is impossible for any one person, location, experience, or method to represent the totality of the Holocaust. This is important because when the Western liberators came face-to-face with such camps as Buchenwald, Bergen-Belsen, and Dachau—and their numerous sub-camps—they were seeing only the very end of the Holocaust. They were seeing it as it was during the final days of the Nazi regime, not during the height of the shootings in the East or the extermination camps in Poland.

Importantly for your question is that they were seeing the end of the “Final Solution” as it blended back into the konzentrationlager (KZ) system of repression. Though the Nazi attempt to exterminate the Jews had long mixed and separated from the KZ system, the two were, throughout Hitler’s rule, linked. Yet, it is the nature of the end that is important here.

Prior to the evacuation of Auschwitz in the spring of 1945, the KZ in Germany primarily held political prisoners, “everyday” criminals—including Paragraph 175 violators, the work-shy, etc. Jews in these camps typically came not as a part of the Final Solution, but through its mixing with the slave labor initiatives. Certainly not all camps follow this generalization, but, as a whole, Jews were the minority in the KZ, if not a relatively small minority.

In fact, even after the long march into Germany over the course of the winter of 1944/45, Jews were not always the majority of incarcerated in the camps. At Kaufering, famously depicted in Band of Brothers episode 9, they were the majority. Nevertheless, at Dachau and its sub-camps, only 22,100 of its 67, 665 were categorized as Jews. Much more numerous were the 43,350 political prisoners. In Buchenwald, roughly 1/5 of those there at liberation were Jews. Thus, you can see why a reporter, if they had no other information about Nazi policy and atrocities, might not stress or might even leave off mentioning Jews as victims. Why they might simply state that they were from all nations and religions of Europe. Jews might not be singular. However, reporters did have more information. They knew that Jews were being singled out for extermination.

UPDATE:

I received a response from the USHMM regarding the query I sent them (BTW, if you ever have a question for them, send it. They are quick to respond and want to make sure there are no questions unaswered!)

From a historian who studies US responses to the Holocaust:

I don't know of such a blatant policy, and I've spent enough time with the US government figuring out liberation and 1945 that I'd like to think I'd have noticed one. In trying to figure out where that sentence might have come from, and I think I have an idea. 

First, the War Department refused to publish a synopsis of the Auschwitz Protocols in the fall of 1944, rejecting it as "too Semitic." But after the War Refugee Board leaked the Protocols to the press and the story was printed nationwide in late November 1944, the army asked for copies of the report for its magazines. (I am pulling that information from the WRB papers; I have not checked Yank or any other soldier newspaper/magazine to see if it was ever excerpted.)

Second, on March 20, 1945, there was a meeting at the Pentagon between the State and War Departments, with a representative from the War Refugee Board sitting in, to finalize the text of one last warning the United States planned to transmit over Germany. I'm not sure if it resulted in leaflets or radio broadcasts, or what the distribution of the statement eventually was.  But in the discussion, the War and State participants (with the exception of George Warren, a refugee expert at State), agreed that the word "Jew" should not be used.  Florence Hodel, the WRB rep, wrote a memo of the meeting and included the sentence "G1 and G2 stated that the word 'Jew' could not be used in the statement." (I've attached the memo. The citation is War Refugee Board papers, Microfilm LM0306, Reel 4, 899-900, USHMM.) 

Now, it's possible that the representatives of G1 and G2 were reflecting a specific policy that I don't know about and can't find specific reference to from the resources available to me right now.  The book I would reference is Joseph Bendersky's The Jewish Threat: Anti-semitic Politics of the US Army, which is partially online and my copy is stuck in the office. There's nothing in the pages available online that indicates a set policy.

We might be able to track down more with access to the library and when NARA opens, but as I said at the beginning, I've never seen a set written policy about the use of the word "Jew." 

Additionally, the historian wanted to point out that...

1. There were some outright anti-semites in the army and State department. Many others believed US soldiers wouldn't fight if they thought they were at war on behalf of the Jews. What we see, however, is that many, upon seeing the camps, wrote home that they now truly understood what they were fighting for. This, though, would be be complicated by the various ways the Nazi atrocities had been represented to them in the past. Did they see various nationalities of people and subsumed Jews under each of these? Were these entirely political persecutions in there minds?

2. Of all main camps liberated by the Americans, only Mauthausen had a prisoner population in which Jews were the majority.

You may also be interested in how /u/kieslowskifan responded to Why were the extermination camps Treblinka Belzec and Sobibor barely mentioned during the Nuremberg trials?