Remember, at that point, there was no "heat of the moment" yet.
As I saw it, T turned to his R, grasped his head, and went down, for no clear reason. Then agents rushed to his side in support.
This could easily have been lightheadedness or worse, and no one would have immediately thought of shots fired at T.
This was live coverage, everyone was scrambling to understand what was going on. I heard it on NPR radio and thought he must have tripped on something.
It was after that that there were suddenly reports of multiple shots and people in the crowd down.
No need for more conspiracy thinking here.