This is a very twisted interpretation. The camps were set up and organized by the RSHA (Reichssicherheitshauptamt = Reich Security Main Office) which was directed by Heinrich Himmler, head of the SS, Gestapo and Minister of Internal Affairs (Head of Police) after '35. The SS ran the camps, and as such the SS was not part of the army. So, German regular soldiers (of those drafted into the regular army, the Wehrmacht), did generally not serve in the camps for this pure logistical reason -- it was not the army's prerogative.
The SS was an all-volunteer organizations until the last stages of the war ('44 or so), when it did began drafting as well. Having said that, it remains one of the shocking aspects of the Holocaust that when Nazi Germany began in '33, the SS counted about 50,000 men, and at the end of the war it counted 1,200,000. Of that latter number, by the spring of '45, more than 600,000 were from countries other than Germany, and most of them volunteers (eg 80k from France and Belgium each, 40k from the Netherlands, 20k from Sweden, 20k from Lithuania etc etc).

The SS had military units fighting in the field alongside the regular army (these were the so-called Waffen-SS) but again, these were not the ones running the camps (but did commit some of the worst atrocities outside of them).

So, of the SS serving in the camps (the so-called Totenkopf-SS, the "Skulls"-SS), the majority was German, but there were often local units and auxiliary police involved. Of the non-Germans serving in the camp, some countries were statistically "overrepresented" (e.g. Austria -- ca 20% of the Auschwitz SS guards were from Austria, which is less than 1/10th the size of Germany by population). So the picture is a little more complicated than that. The overall death apparatus of the Holocaust (camps, guards, execution squads, train logistics etc) roughly required about a million people actively participating, out of which approximately half was German.
SS men after the war would sometimes say they made a choice to be Waffen-SS rather than Totenkopf for moral reasons, but the truth is that within the Nazi apparatus the Waffen-SS was considered of "higher value" or more "heroic" than the lowly camp guards, whose careers usually did not reach much beyond, whereas Waffen-SS officers could rise to the status of a General equivalent. The way to make a career in the Nazi state was less likely through the Camp guard system, which was therefore self-selecting the low intelligence brutes it needed. "Psychologically taxing" had little to do with it.

The concentration camps operated under the direct supervision of the SS, reflecting their steadily increasing dominance over all aspects of policing and internal security in Nazi Germany. The pursuit and persecution of perceived 'enemies of the Reich' and the enforcement of Nazi ideology was the SS' primary mandate.

In recruiting and promotions, the SS prioritised ideological orthodoxy and obedience to the regime. Unlike, for example, the German military, the SS had no institutional identity or heritage separate from Nazism; it was built from the ground up as an organ of the Party, with the expectation that its members' loyalty and commitment to the Nazi cause would be absolute and unquestioning. That ideological steadfastness is why the SS, rather than the military, was entrusted with the security of the Nazi state and the implementation of its racial theory.

The administration and operation of the camp system was the responsibility of the Concentration Camps Inspectorate (Inspektion der Konzentrationslager, or IKL) from 1934-42, and of the consolidated SS Main Economic and Administrative Office (SS-Wirtschafts- und Verwaltungshauptamt, or SS-WVHA) from 1942 onwards. Among other responsibilities, SS-WVHA was the organisation responsible for the selling of slave labour from the concentration camps to private industry, and the seizure and disposal of prisoners' personal property.

Concentration camp guard forces were provided by the 'Death's Head Units' (SS-Totenkopfverbände, or SS-TV), a specialised military formation which emphasised ideological indoctrination and cultivated brutality in its recruits.

SS chief Heinrich Himmler learned relatively quickly that mass executions had a serious impact on the morale and psychological stability of the men carrying them out (observed in even the SS' own Einsatzgruppen, let alone the somewhat less ideologically-indoctrinated regular army.) The industrialisation of the Holocaust, and the adoption of first gas vans and then static gas chambers, was in large part an effort to increase the efficiency of the slaughter, but it was also intended to reduce the psychological stress on its culprits.

In August 1941, Himmler visited occupied Belarus, and observed the Einsatzgruppen firing squads at work. According to his biographer Peter Longerich, Himmler left the front determined to make the annihilation of Europe's Jews more efficient and less 'stressful' for the SS:

It was probably during his visit to Minsk in mid-August or shortly thereafter that Himmler issued his instructions to find a method of killing that exposed his men to less stress than the massacres. [...] Finally, the decision was made in favour of using gas vans. Before the end of the year all four Einsatzgruppen were using this method.

The rapidly-increasing scale and mechanisation of the Holocaust also meant that the SS needed more manpower. From late 1941, the SS began recruiting volunteers from among Soviet prisoners of war to serve in various auxiliary roles, including as combat troops. From among these Hiwis (an abbreviation of Hilfswilliger, meaning 'volunteers'), several thousand were sent for training as auxiliary concentration camp guards by SS-TV at the Trawniki camp in Poland, giving rise to the nickname 'Trawniki men' or 'Trawnikis'.

Comprising predominantly Ukrainian, Belarussian and Baltic recruits, the Trawnikis were not members of the SS; they were a blunt instrument used by SS-TV for some of their most brutal tasks. Ideologically, most weren't Nazis, but they were commonly nationalists and deeply anti-semitic. Overseen by German officers, the Trawnikis quickly gained a reputation as enthusiastic killers; frequently drunk in the field, the SS-TV regarded them as difficult to control, but brutally effective.

The Trawnikis were deployed extensively in Aktion Reinhard, the systematic extermination effort that represented the most intense phase of the Holocaust. The gas chambers at the six dedicated extermination camps — Auschwitz, Bełżec, Chełmno, Majdanek, Sobibór and Treblinka — were operated by Trawnikis. The Sonderkommandos (groups of prisoners forced to dispose of the bodies of the camp victims) were also overseen by Trawnikis.

Trawnikis were also heavily involved in the massacres that accompanied the liquidation of ghettos across occupied eastern Europe. In Warsaw, for example, Trawnikis made up around 15% of the forces deployed to suppress the 1943 Ghetto Uprising and subsequently liquidate the ghetto.

Although an unreliable and undisciplined force — desertions and insubordination became increasingly common through 1943, resulting in the reassignment of Trawniki units away from several of the extermination camps to other sites — the Trawnikis nonetheless played a crucial role in the deadliest phase of the Nazi genocide. Peter Black, the author of one of the relatively few detailed studies of the Trawnikis, writes:

For all their indiscipline, for all the harsh treatment they received from their German trainers, and for all the desertions, the Trawniki men helped make it possible for [Aktion Reinhard overseer Odilo] Globocnik, with his tiny staff of fewer than 200 men, to carry out the physical elimination of 1.7 million people. This dreadful accomplishment required the support of an engaged and ruthlessly led auxiliary force, inspired in part by a leadership that accepted them as valuable—if not necessarily equal.

[...]

The Trawniki-trained guards were necessary to the implementation of Operation Reinhard: they inflicted some 28 percent of the human loss sustained by the European Jews during World War II.

The role of the Trawniki men has come under more intense scrutiny in the past 20-30 years, in large part thanks to the widely-publicised extraditions and prosecutions of several non-Germans who served in the concentration camps — most recently, the case of John Demjanjuk, a Ukrainian who served at Sobibór.

The camps were run by the SS. To be specific, they were run by the WVHA, the SS economic office. This was because part of the camps purpose was to dispossess and reprocess property taken from Jews. A lot of the Camps also contributed to the economy by providing Slave Labor to both the German war effort and Private businesses that wanted to use them.

An exception would be the "pure" extermination camps of Action Reinhard, Belzec, Sobibor and Treblinka. They coordinated with the WVHA in processing stolen property from Jews, but were founded by and under the Jurisdiction of Odilo Globocnik, the SS and Police Leader in The Lublin District of occupied Poland, known as the "General Government. In other words, these camps were under the RSHA, the SS Reich Main security office, which by the name handled all "Security" in the Nazi German Reich, and also controlled all German Police.

In General, the SS recruited collaborators/Volunteers from Eastern Europe. Specifically Ukrainians, Lithuanians and other Balts. These were taken from Soviet POWs, which was because they had no loyalty to the USSR and in many cases shared the Nazis' Jew hatred. The Reinhard camp staff had several Eastern European collaborators, the most famous being "Ivan"/John Demjanjuk, who was a guard at the Sobibor Extermination camp. The SS "Judenvernichtungsbatallion" - Aka the Squads that went from town to town In Nazi Occupied Poland and deported Jews to the extermination camps, were also made up of Collaborators, under the command of Nazi German SS Officers Adolf Feucht and Erich Kapke.

You can find a source for Ukrainian/Baltic/Eastern European collaboration with the Nazis, from the work of Christopher Browning. For example, Ordinary Men.