As even you agree the game clearly starts and stops when either the ball is out of bounds or a player is hurt. Therefore it would make sense to stop the clock.

Did you really mean that the palyers and fans have no idea how much time is left in the game? That seems truly bizarre. In other words the players, coaches, etc. don't even know that the end of the game is approaching and they need to hurry and try to even the score and no one knows how much extra time will be added, how can the players play like that? It removes any end game strategy and tactics.

you wrote: "What is the objective difference between stopping the clock, and adding the time on at the end?"

For one, the additional time is only an approximation of the time lost as it is in whole minutes. If you stop the clock you don't have to approximate anything, the clock is exact. Secondly, it just seems much simpler to stop the clock. It seems very kludgy to have the clock keep running and then add on time.