It sounds fine to me. I would prefer the phrase "last stop" or "final stop" just because "final destination" reminds me of the movies (I think they're movies... I've never actually seen them). But either wording is fine.
"Final destination," to me, means where I ultimately want to go on this trip: my home, a friend's home, an office, a store. My destination is unlikely to be a subway stop, even one at the end of a line. I am almost certain to continue my trip by bus, by taxi, by some other means of transportation, or on foot.
I have heard this phrase used in a similar way on flights, where a cabin crew member, on welcoming us to an airport, often says something like "If Chicago is your final destination, you may collect your checked baggage on Carousel 3." (Presumably some passengers will connect to other flights.) It's a bit softer than the usage you cite because it considers all of Chicago, and presumably its surroundings, to be part of the destination - not just the airport. Still, it's similar, so perhaps the usage you found isn't that bad.