Number 1 Rule of Action Shows...
...is that you never shoot the tires.
"Shoot the tires!" we shout if it's Jack doing the shooting. The bad guys labor under the same handicap. (Must be some sort of union thing. But then that makes sense since you couldn't have the obligatory chase/escape scene if you had cars getting grounded all the time.)
So Jack steals a policecar since the two policemen are, shall we say, indisposed. But fortunately the bad guys honor the Marquis of Queensberry "Never Shoot the Tires" rules.
The other imponderable is how when Jack received the recording implicating President Logan, he didn't immediately called Chloe and have her record it. That way if he gets killed the message lives on. Go figure.