The one that gets up my nose is '...because of the late arrival of the incoming aircraft', which is just a roundabout way of saying 'it's late because it's late'
No not even a roundabout way, it is a direct way of saying just that. This isn't meant to insult your intelligence, but look at it from the pilots viewpoint. When we make a PA on stand prior to pushback, it is often under a severe time constraint. Usually we are pushing an imminent departure slot that if we fail to make, may result in additional delays, and sometimes substantial delays. Very often we do not have the detailed history of what happened to the aircraft yesterday, or overnight, that caused it to be consequentially late for this departure. Even if we do, the detail is often of precious little interest to most of the passengers, who are more concerned about getting on with the journey they have contracted for. It is often a case of people consuming the sausage rather than wanting to know the detail of how it was made. So the reason for the delay is the late arrival of the inbound aircraft. This differentiates the cause from mechanical dealys, ATC slot delays, Loading delays, Late arriving passengers and other security delays etc. Any of those things might have happened yesterday, but the majority of people (including the crew) simply want to know when the flight will depart and their delay will end.
On top of this explanation (however brief) there are welcomes, routine information and safety reminders to incorporate into the minute or two that might be available. Communication is important, and most people appreciate clear, informative and positive communication from the flightdeck. In many cases it might be the first straightforward explanation they have been given concerning an irregularity.
There are rare occaisions when some oddity or bizzare event has caused a delay, and there may be a degree of humour that can be extracted from the situation, but most of the time it is fairly mundane stuff, and of course the late arrival of the inbound aircraft.