Be it legitimate or not, as far as I'm concerned it's totally pointless as an announcement. The point is that it's delayed and that's all I need to know. Obviously it's delayed for one of a myriad of reasons which will have affected that aircraft that day (or even the previous day in some cases). To tell me it will be late departing because it's late arriving is quite frankly taking the pi$$! Just announce the delay and try and be as accurate as possible about the new departure time.
It's perfectly legitimate. It's telling you the aircraft isn't there yet. Most other delays involved the aircraft being at the Airport and another issue with it. Start giving reasons for previous delays and you can have a long list of delays throughout the day for the reason the last flight of the day is late. Pointless long announcement that will just cause more confusion.
Also most of the time the ground agents don't know the reason for the previous delays. Airline Operations will advise the delay reason via IATA Delay codes and often only the reason for the delay to that particular flight