Because today's Silver Crown cars have lineage going back to the Championship cars that raced in the Indianapolis 500 from the start. The rules always called for transmissions and starters (whether on board or external to save weight) and required pit stops. Many of the same cars that ran in the Indy 500 also ran on the mile dirt tracks. Push starts were specifically not allowed. Lower classes of racing allowed pull or push starts and only ran for very short distances. For decades up through the 1950s Championship cars were even required to have a reverse gear with the idea being that a driver could use reverse to back up and get out of the way after a spin if needed.

Today you probably could do away with transmissions and push start the cars but all you would have would be oversized and under powered sprint cars.