Maybe not in your state. But, it does in mine. Shooting out tires to "prevent reckless driving" is the same as doing so to "prevent some from leaving".
And you missed my point: laws governing use of force in some states are the same for both law enforcement and non-law enforcement.
Further, "shooting out the tires" is not accepted police procedure in some departments. Even though it is legal to shoot at a fleeing felon under certain circumstances, department policy prevents it. Your state may be different. Your city may be different. And since the Park Police is a federal agency, their rules may be different as well.
You are so caught up in "what they should have known", that you can't stop and think: "what did they know"?
I'll say it again: this whole incident would have ended peacefully if the driver had cooperated with police at any one of the three places she was stopped. Rather than asking "what should the police have done?", you should be asking: "what should the driver have done?".