I've never seen an airport where there was pro-active information that bags didn't made it right at the luggage belt, unless like half of the aircraft was missing theirs. Probably there are some airports that have some kind of setting of individual pre-information, but I've yet to experience it.

Also, usually lost&found baggage office is right next to the luggage belts. Isn't that the case in JFK? Looks like a weird setup if you've to get landside for the lost&found office.

When someone else took my bag in PEK, the proper CA lost & found staff was very friendly and helped me to reunite me with my bag in an efficient manner (and the other pax with his) - not much Air China can do at some random outstation to improve service there (it's often not even airline staff at outstations, but rather 3rd party personal)