The reasons are twofold. A. the lack of manpower Germany had, and B. that Nazis were not above using Eastern Europeans to do work they did not want to do, including mass killings. In regards to mass killings, the psychological effect on the German soldier was severe, and Germans would resort to many forms of reducing the effect, including the use of locally formed militias/groups.
I recommend checking out Ordinary Men: Reserve Police Battalion 101 and the Final Solution in Poland by Christopher Browning, as it is a psychological study on the German Police Battalion 101, and includes stories about the use of the militias in mass killings.
I also recommend checking out Stalingrad by Anthony Beevor. He talks as much as he can about the Ukrainian troops used in the German army as much as he can. However, there is a lack of documentation on these units and individual soldiers, especially due to the restricted access to Russian archives.
Let me use Ukraine as an example, as it is indicative of how many in the other countries occupied by Stalin and the USSR felt as-well. Initially, many Ukrainians welcomed the German army with open arms, giving them food and cheers when advancing into a village. They thought the Germans had actually come to liberate them, and the Nationalists in Ukraine quickly moved to support both the SS and the Wehrmacht. Local militias and volunteers were formed, however, I do not think they were ever allowed to join the SS due to their status as sub-human, maybe another person can clear that up for me. I do know they accepted individual volunteers into the Wehrmacht, especially as the Battle of Stalingrad./ Operation Blue drug on and manpower became short. Also, the SS-would use militias to do mass killings and other types of security work, but were not generally formed as an actual part of the SS. The Germans never received any real reinforcements from Germany, and so used many foreign volunteers and other foreign divisions in the Battle for Stalingrad, whereas before they were not really trusted to be in combat. However, to the Ukrainians, it was eventually realized that the Germans came as subjucators not liberators.
As stated before, Germany was woefully short of mobilized manpower (as opposed to the vast amount of immobilized women), so they were desperate for foreign slave labor. Up to a million persons working in Germany's arms industry were slave laborers from around the conquered territories, including much of the USSR by the end of 1944. That does not include all of the unrecorded forced labor at the front lines, which was most certainly used on a massive scale. Subjucation of the Slavs for the use of Germans was set out in Mein Kampf. Moreover, after it became clear that the Germans were not here to liberate Ukraine, the volunteers really had no choice but to do what the Germans asked of them, being so closed to the SS and German army. Either A. be killed by the Gestapo for refusing or B. be killed by the Soviets for betraying the motherland. Stalingrad depicts this in personal accounts, how commradery was formed in Stalingrad between the coerced or volunteer Ukrainians in the German military and the Germans fighting in the city. The NKVD detachments, and Russian military soldiers would interrogate and kill the Ukrainians when captured. They had little choice but to fight till the end. Running away to the German occupied steppes in winter was not really an option. Besides that, the Nationalist militias where overseen by the SS Einsatzgruppen, and kept on a short leash by professional secret police. Moreover, they were usually sent to do mass killings against other ethnicities, especially ones they hated.
The psychological effect of mass killings were debilitating on most German soldiers. Many ways were used to dull the effect of killing helpless civilians on the mind. Alcohol was common during or after a mass killing. Even the way Germans executed people was designed to have the least effect on the mind. Laying people down, using a bayonet to aim (so as not to have to see the victim), and shooting the back of the neck (to avoid blood and gore splatter). The most effective method used however, was personally removing the soldiers from directly engaging in the final execution, rather than indirectly doing it. They corralled what they thought of as sub humans to kill other sub humans, and also used the death camps for the aim of genocide. Even though the Police Battalion 101 would escort a Lithuanian militia to the mass execution, thus ensuring the deaths of the civilians, the psychological effect on the German soldier was far less. The Germans knew about the trauma caused to the soldiers, and would not even persecute many from returning home instead of participating in mass killings, as shown in a speech made by Goebbels and personal accounts. "Go home and collect your pensions" Goebbels said in his speech. Using these slave laborers to work and militia/coerced groups to commit mass killing directly coincided with Nazi beliefs set out in Mein Kampf, but the Wehrmacht was certainly not against using any manpower they could get.
The SS and Wehrmacht were largely independent organizations, with Himmler controlling the SS and and exception being Hitler giving personal commands to the Wehrmacht to help the SS. The Wehrmacht was much more focused on obtaining victory and less focused on ideology, whereas the SS were born out of Nazi ideology. It is no surprise the army used any manpower that they could get, only citing practical reasons of competence when not using it, even though that reason may be based on racist views. However, the military would overtime become more filled with fanatical Nazi's and less filled with competent commanders. However, I have not found any concern or even knowledge from Hitler about Untermenschen serving in the military, especially after the lack of reinforcements and subsequent use of foreign divisions in defensive roles. Hiwi units (Ukrainian units) were regularly used in the meat grinder of Stalingrad. I will admit I do not have much knowledge on what Hitler thought of these men serving in the military, so if someone could clear that up for me.