First, assault guns came into their own for a number of reasons. Kramer, as far as the silhouette, you are right, it was very low with the Stug III, Stug IV (Guderian's Duck) and the Hertzer. It was a little higher with the JadgPanther. The point I was making about the chassis of the Stug III and Hertzer (translation: "Troublemaker") is that the tank turrents that previously resided on these chassis could not support a capable anti-tank gun, or thick sloped armor. This became terribly obvious when the 6th Panzer Division with about 150 long-barrel 50MM Panzer III's (and a small number of effective long barrel 75MM Pz IV's) led the relief effort to break the Stalingrad ring, with the help of the seriously understrength 11th panzer division, made up of smaller numbers of primarily of Panzer III's. When they came up against very large numbers of T-34's in long range engagements near the Aksai River, they suffered very very high casualties. On December 17, 1942, the primary battle group in the relief effort lost 90 of their 120 available panzers (although some were repaired and fought again). This basically ended any realistic opportunity of freeing the 6th Army. Based on these experiences, the report made by the 11th Panzer division on January 22, 1943 regarding the Panzer III was:
"The Panzer III is in no way equal to the demands of the War in the East. Its armor is too thin, the calibre of the gun is inadequate. In contrast, the assault gun has proved to be an outstanding success in the Steppe war, even though it lacks a movable turret. The reason for it's success lays in it's considerably stronger armor and more powerful gun."
After this report, Panzer III's were quickly phased out and all production capability was redirected towards Stugs. 2150 50MM Panzer III's were produced in 1942. Only 22 of these tanks were made in 1943, although about 210 short-barreled 75MM Panzer III's were also produced. Meanwhile, Stug III production increased from 700 in 1942 to over 3000 in 1943.
As Roger points out, the assault gun began it's life as an infantry support weapon and was well suited for this. However, it's adaptability of carrying a higher calibre gun than a turreted tank on the same chassis, along with it's low silhouette and heavier armor (which was also sloped, making it more effective) than it's tank eqivilant, made it an ideal anti-tank weapon. It was also relatively cheap and easy to build in existing production facilities with little retooling.
As far as Stug III's and Hertzers being obsolete in two years, that is mostly correct. However up until early 1945 when the first 90mm American tanks began arriving, the L48 gun was as good or better than anything the Western allies had. Only the Sherman firefly (which was available in small numbers) could compare with it. It could also take out a T-34 fairly reliably at ranges inside of 1000M, and Shermans at even longer ranges. In other words as long as T-34s and Shermans were on the battlefield, they would still be somewhat effective. The JadgPanther and JadgTiger were responses for the need of a more powerful anti-tank oriented assault gun to take out heavier tanks. The JadgPanther was in most expert's eyes the best tank destroyer of the War. It's 88mm L71 gun was a more powerful version of the Tiger I's 88mm gun, and was extremely lethal, along with having extremely thick frontal armor. Unfortunately it's factory was ravaged by allied bombers and only about 400 of these could be produced.
Hope this clarifies things.