I would like to have lunch here on the 9th December following our carriage ride in Central. Park. About 1.30. I can't get a slot then. Does that mean we can just arrive and wait. Or there's no chance of a table.

Check their website - did not think they take any reservations. If really nice weather (low probability) there would be a wait... if not amazing weather likely little to no wait

Sounds strange, we've called and booked many times for dinner

They'll be open for lunch until 15:45, you can access bookings through their website or OpenTable. We arrived ala 'walk in' the other week & only had to wait 15/20mins for a table about the same time.