Rule Question

At the risk of sounding very stupid: I watched the UT/Huskies game without the audio commentary. With 50 seconds left I was expecting the game and play clock to start simultaneously after the Washington player was injured. So I thought the ball would be punted with 25 seconds or so on the clock. My assumption about the rule was incorrect as the game clock did not start until the ball was snapped. Would the same rule have applied if the injured player had been a Longhorn? In all of my years of watching football I can't remember seeing this exact situation on a 4th down like this before.