One note about the phrasing of your question, is that "war machine" is a metaphor for a nation's entire military system (particularly in the age of total war) - it doesn't literally refer to a mechanised army. You could describe Hannibal's Army as part of Carthage's war machine, for example.

Strength in war is relative and almost all World War Two armies relied on railroads for logistics. In 1939-41 the German army was the most logistically sophisticated army in the world. In 1942-45 it was the second most logistically sophisticated army behind the Anglo-American led western allied forces.

For most of the war, for most armies, the main logistical arteries were railroads by land and ships by sea and river. The last few miles between the front and the railhead or port were crossed by horses or mule drawn wagons. Motorized trucks were secondary for the most part, with the key exception of armored or mechanized units which used them extensively. Non-motorized Infantry divisions made do with far more limited access to motor vehicles. This system was basically used by all armies, even occasionally including the Americans and definitely by both German and Soviet armies in the eastern front.

The only large force that really broke this mold was the Allied forces in France and Germany in 1944. They had enough motor vehicles to mostly avoid using mules or horses for the "last mile" between the railhead (or port) and the front line. Additionally, they could cope with destroyed railroads by using huge numbers of trucks to replace rail transit and restore long range supplies and mobility... barely. Allied logistics broke down during the rapid advance in 1944 and leading units had to curtain their advance due to lack of supplies, just in the same way that German units had to curtail their advances in 1941 and 1942 while the railroads caught up to the front in the Soviet Union. Additionally, where roads didn't support mechanization, like in mountainous areas of Italy, western allied forces also used mule drawn logistics.

While the Red army used quite a lot of trucks, a portion of them acquired by Lend-Lease from the US, their advances were also characterized by long pauses to allow railroad based logistics to catch up to the front. For example the center of the Soviet offensive in 1944 was paused from August to January in order for logistics (meaning railroads) to catch up to the fighting front.

American use of trucks in 1944 was the first instance of a large national army phasing out horse-drawn supply and cutting the tie between military operations and the railroad network. It doesn't make sense to harshly criticize the German army by comparing it's operations to the Americans in the last year of the war.