they're only "entitled" to LPR if CIC says they are.

My point about the Canadian driving a US car across the border was a reversal situation. If a Canadian with visitor status in the US attempts to drive a US registered car across the border into Canada, Canadian border officials won't allow it. They'll impound the car and return it to the US.
If a Canadian citizen who is a US LPR however is driving a US registered car across the border, they'd waive them thru. Unless of course the car is not registered to the driver, or any of the passengers, then they might have a problem, if Border officials happen to check.