It came too late to have any major impact on the war. Development on what would ultimately become the StG 44 started in early/mid-1942. By the time any meaningful numbers of these weapons could be deployed, the Germans had already lost the strategic initiative on the Eastern Front.
I'll agree that technologically, the StG 44 was leaps and bounds over the Karabiner 98 kurz (K98k). The K98k used the older 7.92x57 (known as 8mm Mauser) cartridge, a holdover from World War 1, while the StG44 used the newer 7.92x33 (known as 8mm Kurz). With a larger cartridge, the K98k had a higher muzzle energy than the StG 44, which is good for combat at longer ranges, but generates larger recoil. In a bolt action system, this isn't really a problem as there is time between shots. However, in a semi-automatic and full-automatic system, the recoil can affect subsequent shots. In urban/close quarters, rate of fire and suppression are more useful than effective range. The Soviets had deployed large amounts of PPShs and SVT-40s, giving them a tactical advantage over the Germans, who were still largely equipped with K98ks. Entire Soviet infantry companies were armed with PPShs as opposed to the Germans, who mainly were armed with K98ks, with the exception of squad and platoon leaders, who would of maybe had an MP40.
The jump from K98k to StG44 isn't correct either. Germany had developed semi-automatic firearms in the for of the Gewehr 41 (G41), Fallschirmjager Gewehr 42 (FG42), and the Gewehr 43 (G43). The G41 was a semi-automatic, fixed magazine rifle that used the 8x57 cartridge. However, the design was poor, as the Heer (the German Army) placed design restrictions, limiting the design's effectiveness. The FG42 was a select-fire, magazine fed rifle that used the 8x57 cartridge. The Luftwaffe commissioned the design for it's paratroopers, the Fallschirmjagers, but the design never really took off as the Heer controlled most of the military factories and production output. The G43 was a semi-automatic, magazine fed rifle chambered in 8x57. The design was similar to the SVT-40 and was more reliable than the G41.
Finally, a modicum of understanding about the German wartime industry is needed. The Heer/OKH controlled most of the wartime industry. There was a lot of politicizing in the German wartime industry. The Heer refused to arm the SS-VT/Waffen-SS, as they weren't officially part of the Heer. They had to enter into a contract with Steyr directly in order to obtain arms before Hitler issued a decree ordering the Heer to arm the SS. That being said, the Heer also dictated what weapons were being produced. Most of the plants were tooled to manufacture K98ks and MG34/42s. The German infantry doctrine at the time was still based around the 2 machine guns in the squad with the rest of the squad being riflemen and supporting the machine guns. It also takes time and and money to acquire new tooling to produce different weapons. This was evident in at Waffenwerks Brunn 1 and 2 (plants in Czechoslovakia obtained after the invasion). For the first year or two under their operation for Germany, they only produced weapons which they had produced before the invasion, albeit with slight German modifications. Finally, the Germans didn't adopt a war economy until 1942/43.