As far as I can see, the rule simply says same-day DL or DL-partner flight. I don't see any reason you can't use it on arrival. https://www.delta.com/us/en/delta-sky-club/access

If you do decide to abandon the last leg of an itinerary, the general consensus is that you should just quietly no-show without saying anything. Trying to cancel to avoid stressing the gate agent, while very well intentioned, creates more problems and raises suspicion. Flat-tire rebooking is a terrible idea, forget about that completely. In any case, make sure you don't have upcoming return flights on the same itinerary. As soon as you skip one leg, all future legs are cancelled