Another possibility is that rentals paid with points through the Chase portal are actually paid with the Chase card, and the points are used to offset the card transaction. Does a charge/credit appear on the card?
Not as far as I remember. It's been awhile since I've booked a car with pts but I'll say "no" if you book 100% of the cost with pts.
I will not take a chance and only use a c.c. for my car rentals.
Activities are a good use of pts however. I'm going to book snorkel cruises & whale watching trips with mine. a 50% savings is pretty sweet. The balance will eventually be xferred to airlines & hotels.