Personally I have always liked the PA from the flight deck (and the cabin crew too, I always pay attention (even though I have heard it hundreds of times. My background for many years was always safety related in various ways with checking and reporting systems in place)

Although I am not a nervous passenger in any way I still find it reassuring but most of all the 2 or 3 Pilots up front have a huge responsibility to get us from A to B safely. On short haul even a very brief intro is welcome although I do understand on turnaorunds they probably can have lots of other flight deck duties they need to do, unexpected things happen etc etc.

PA announcements of course vary and it's interesting sometimes to hear what route we will be taking, any anticipated weather issues as well as the important anticipated flight time.

Finally I think it's also adding the human touch, it's very important in my view, to hear that calm introduction and as others have pointed out it can have a diluting effect on passengers nervousness. Long may this last part continue please.