Germany's arms manufacturing industry was pretty much dismantled by the Allies during the occupation of Germany, as a part of a much larger effort to end Germany's war-making potential. It wasn't just the arms industry - factories and facilities of all kinds were dismantled and destroyed by the Allies (both the Western powers and the USSR), with equipment frequently being carted out of Germany as a form of reparations.

German arms continued to be used in certain circumstances, such as by the new German police forces that were formed in the West and East, but for quite a while there was basically no arms manufacturing going on in Germany. The German arms industry would obviously recover, with firearms being produced in both the West and East during the Cold War, but by that point the StG 44 was not particularly relevant. In East Germany, the arms industry centred mainly around making the same sort of weapons that the Soviets and the other Warsaw Pact nations were using, not indigenous designs. The National People's Army and the Volkspolizei definitely used weapons built during the war, but as time went on they mainly adopted firearms that were standard throughout the USSR. In the West, Walther reformed pretty quickly after the war, but they mainly focused on manufacturing pistols for the army and police (handguns being Walther's main area of expertise). The Western German Bundeswehr, formed in 1955, issued a call for a new service rifle in 1956. The contract was won by Heckler & Koch (a post-war company formed by former Mauser engineers) with their G3 battle rifle design, which fired the 7.62×51mm NATO round. Using the same round as other NATO battle rifles, like the famous FN FAL or the American M-14, obviously was preferential to West Germany maintaining their own unique calibre.

At the time, West German doctrine, and NATO doctrine in general, favoured powerful battle rifles that fired the full-size 7.62×51mm NATO. The StG 44 was an assault rifle and it's round, the 7.92×33mm Kurz, was an intermediate round that was much less powerful than the 7.62×51mm NATO round. The Soviets were quicker to recognize the value of assault rifles (namely, their own Kalashnikov design) than NATO was, and by the time NATO nations began looking at replacing battle rifles with assault rifles, the StG 44 was thoroughly outdated.

Basically, in the West, the assault rifle was not seen as the ideal standard service weapon until much later on in the Cold War, by which point the StG 44 was outdated. In the East, the value of the assault rifle was recognized early on but the arms industry mainly focused on manufacturing Soviet/Warsaw Pact designs as opposed to developing indigenous weapons. The AK-47 was mostly a superior gun to the StG 44 anyways, and it came along not to long after the war, in the late 40s, before German arms manufacturing had recovered.