Wellsir, since every state is sovereign, at least until the people of this country give away their freedon, which reminds me of a line from one of the "Star Wars" movies: "So this is how the republic dies; to thunderous applause..." but I digress.
Anyway, for now we are a union of Sovereign states, each with a constitution and it's own laws. And I can only respond to your comment based on Jawja Law.

Just for the sake of argument don't cha know...
Their excuse is it's private property and they have no jurisdiction.
You'll hear the same thing here. What they really mean is that traffic laws do not apply to private property. So they can't give anyone a ticket. Some agencies may do a report out of courtesy, others may not. Atlanta used to give the parties a state report of accident form, an SR13, to fill out themselves. BTW, some things that people consider traffic offenses, like DUI, really are not and the private property thing doesn't apply.
Wonder how that would work for you if you killed someone in that parking lot or your home? Could you deny them entry because it's "private property"?
Good luck tryin that!

Actually, you may be able to deny the police entry onto your property, depending on the circumstances of the instant situation, but that doesn't mean you didn't violate a law against taking a life, if there is such a law in your jurisdiction.
The state police
Now we don't got no state police here in Jawja. We have a "State Patrol" They actually would NOT have jurisdiction on private property or most streets here in Jawja. If we had a State Police, they would have the same jurisdiction and limitations that the local police would in a private property accident scenario within the local LE agency's jurisdiction.
have a different outlook about accidents in parking lots. According to them if it is there for public use it doesn't fall under the definition of private property and they will respond to accidents, investigate, and issue citations.
This is true in certain, legally defined, situations. And just being there for public use, like a mall parking lot, is NOT one of those situations in Jawja. Again, application of the definition is the same for local or state agencies.
Finally, in Jawja, the state can suspend a persons driver's license and registration if they let their insurance lapse. And the Ins. Companies are required to inform the state if anyone does that. But, in this day and age, uninsured motorist coverage is a necessity! IMNSHO!!!
And thanks O&S, for giving me the opportunity to sit here and pontificate!

It makes me feel good about myself...

'course that bit about traffic laws not applying to private property could use some more discussion... but, maybe I'd better