If they do stop on the way to CSS the place usually has jerk food. Whether they stop or not depends on you and the other passengers. With the new road completed many people prefer not to stop and get to CSS as soon as possible!
We have never stopped going from the airport to CSS, CTI or CSA- and never wanted to. By the time you land, go through Baggage Claim, Customs, wait in the Couples Lounge ( 5-20 min), and drive 2 hours, it's great to just get to your hotel..