This is a thorough myth, which has been promoted since the end of WWII, basically called the "Clean Wehrmacht." But the reality is quite more bleak and actually the German military was just as "Nazified" as the rest of Germany during WWII.

The OKW, the Wehrmacht high command, was thoroughly run by Nazis following Hitler's genocidal agenda, as we have known for years.

The German army was a regular part of the committed genocides in various nations, such as Poland, and committed the mass atrocities against the Russian, Ukrainian, Czechoslovakian, etc. peoples, so as to make lebensraum, i.e. the living space Hitler perceived as being needed for the furtherance of the Aryan race. Even more so, we know that average German soldiers were most certainly complicit in the continuation and furtherance of the Holocaust.

Beorn (see Waitman Beorn, Marching into Darkness [London: Harvard University Press, 2014]), goes into depth about how the German soldiery went from just basic murders of Jewish minority populations, to actively creating games of annihilating them. Likewise, they were a part of the Hunger Plan, to try and starve the Soviet Union into submission, and more.

The German army was just as Nazified as the rest of Germany. There may have been exceptions, like the 20 July Assassination plot conspirators, but they remained exceptions which proved the rule, and it should be noted that many of those figures were far from anti-Nazi, and more anti-Hitler. Claus Von Stauffenberg, as noted by his contemporary who knew him, Hans Gisevius, was just as totalitarian and antisemitic. By all counts, there is little reason to think that Stauffenberg was anything but fed up with Hitler's mismanagement and complete ineptitude during the war (see Hans Gisevius, To the bitter end [Boston: Houghton Mifflin Company, 1947]).

The reasons for their acceptance of totalitarianism are likely those equivalent to why we see many alt-right and totalitarian movements sprouting up in the Americas and Europe today. From an economic standpoint, as Robert Paxton points out, capitalism and fascism may not have always been the most friendly, but they made good bedfellows, see Robert O. Paxton, The Anatomy of Fascism (New York: Vintage Books, 2004), 208. Likewise, fascists have a tendency to unify around nationalist rhetoric which strikes home, and also is very appealing to conservative values about women, children, traditionalism, and more, see Jason Stanley, How Fascism Works: The Politics of Us and Them (New York: Random House, 2018). Additionally, the mythmaking process, i.e. that process by which fascists distort or fabricate a mythic past for their nation and its selected "true" people (in Nazi Germany's case, Aryan Germans), becomes very easily believed and swallowed, especially when it becomes state propaganda used both inside and outside of schools in an aggressive strategy to reach all, see George S. Williamson, The Longing for Myth in Germany: Religion and Aesthetic Culture from Romanticism to Nietzsche (Chicago: University of Chicago Press, 2004).

So... in short, the German army was just as Nazified as the rest of Germany. The myth of the "Clean Wehrmacht" is nothing but a pseudohistory which is promoted in an attempt to save the face of the German military, an effort which was largely run by ex-Nazi officers and ex-Nazi scholars, trying to retroactively redeem themselves and try to reintegrate into society, where they could once again begin fascist endeavors. It is a quite real attempt at mythmaking Germany's past once again, as noted above.