~The Germans did kill POWs, specially Red Army POWs. The US and UK and USSR did the same thing to Axis prisoners. It is a brutal fact of war that surrendering enemies are summarily killed, perhaps out of hatred or vengeance or perhaps because those doing the capturing are incapable for any number of reasons to process and hold prisoners. However, only a fool would doubt the existence of the Extermination Camps. That being said, most people have no clue as to too much about those camps. For example, there were only a dozen or so Death Camps and they only operated for about 18 months, from December 1941 until the spring of 1943. Realizing that the Red Army was winning the war and that the Death Camps were wasting a valuable and necessary source of slave labor, Himmler himself ordered them to terminate operations and to be fully or partially dismantled. It is estimated that as many as 6 million people were murdered in the Extermination Camps. Only about 2.4 million of them were Jews.
There were thousands of Concentration Camps. The KZs were not killing factories. The horrendous death rate was the result of disease, starvation, overwork and exhaustion. Raging epidemics could not be contained due to the overcrowding and lack of medical supplies, and accounted for the bulk of KZ deaths. It is estimated that as many as 18 million died in the KZs, including some 3.6 million Jews. Jews represent a minority of the victims. Of the 6 million Jews who died in both type camps, a significant percentage would have suffered the same fate for other reasons, even if they had not been Jewish. But for the Red Army repelling Barbarossa and defeating the Third Reich virtually by itself, with little significant military assistance from the western allies, NSDAP "racial purity" and "ethnic cleansing" programs had doomed another 30 million ethnic Poles, 30 million Slavs and at least 30 million more Russians and Soviet Communists to extermination. As it was, the genocide against the Serbs, Roma and Sinti, as a factor of population, was more complete than that of the Jews. Not a single operating Death Camp was liberated by either the Western Allies or the Red Army. They had ceased operations long before liberation. Concentration Camps continued operations until the end of the war and were liberated, by the hundreds if not thousands. Auschwitz I and Auschwitz III were liberated by the Red Army. Auschwitz II, Birkenau, was the last Death Camp to halt executions. It too was liberated by the Soviets, but not until many months after the last genocidal execution.
The gas chambers were installed for reasons of efficiency and cost. Not all the Death Camps used them, or used them exclusively. Bullets continued to used even after Zyklon B was introduced. Long before Chelmno opened for business on December 8, 1941, Himmler's Einsatzgruppen had been following the Heer Army and rounding up and shooting captured "undesirables". Himmler and Heydrich decided that not only was the practice a waste of bullets, but it had a detrimental effect on the SS troops who pulled the triggers. Clearly Hitler must have approved of both the Death Camps and the mass executions by the Einsatzgruppen, but there is no proof that he was aware of what Himmler, Heydrich and, probably, Goering were doing until after to operations were well underway. The executions had been going on for almost a year before the Wannsee Conference, which was held to let the second echelon of the German High Command know what was happening and was not held to plan "The Final Solution to the Jewish Question" (a small segment of the overall racial scheme).
None of that addresses your question. Your question, however, indicates that you lack a basic understanding of the camps themselves and the genocide issues generally. There is irrefutable proof that US POWs were sent to both Concentration and Death Camps. Not many, but a few. Since the first US troops to see action did come under fire until November 1942, in North Africa against first the Vichy French then the Italians, not many were captured until after Himmler had ordered the halt to the Death Camp Executions. It was far easier to simply shoot POWs in the field at the point of capture, just as the Americans did with captured Germans. Those sent to the camps were generally suspected of espionage, sabotage or had presented disciplinary problems in POW camps. Most Jewish American POWs were held in POW Camps and the POWs that found themselves in the KZs and Death Camps did so for a reason. Hundreds of US MIAs never were accounted for. Some may have deserted. Some may have died in action and never been found. Some may have died in the camps.Some may have fallen into Soviet hands as KZs and POW camps in the East were liberated. For political reasons, it is more than possible that Stalin refused to acknowledge or repatriate "liberated" US POWs.