Trommelfeuer, you are under a misconception. When you quote "They were used in support of the infantry" that in no way implies indirect fire. The Sturmgeschütz was the successor to the light artillery in WW1, which employed infantry guns in a direct fire role. Obviously, that approach had shortcomings, and putting said IG on a chassis solved those problems.
Principally, StuGs were able of indirect fire, but rarely used in that role. I recall one mention somewhere where Panzer IV (short) were used in an artillery role (they also had special equipment for that), too, but it was uncommon to say the least. The traditional role of the StuG was infantry support, hang back while the infantry attacks, and when they run into a MG nest or bunker or something else that gives them trouble, bring forward the StuG and get it over with
Also, yes, StuG crews were artillerymen, not just early in the war, but there's no rule that artillerymen may not fire on a target they can see
apex