A few years ago I heard a private pilot with an alternator failure going to the same airport I was. Every frequency change he was having a long discussion with the controller making sure they were aware of the issue and that he might lose comms. I remember thinking at the time that a quick pan pan prefix might have reduced the likelihood of having the comm failure he was worried about. The prefix can certainly be appropriate when you want to make sure everyone is on the same page.
On a similar note, I attended a lecture by Tammie Jo Shultz and Darren Ellisor a couple years ago talking about SouthWest 1380. They had something like 14 frequency changes and were asked 3 separate times for all the information, including fuel on board in pounds, then hours and minutes, then pounds again. When asked why they didn't just use 121.5 and tell the controllers to deal with it they said they had asked for it and been denied. Mr Ellisor pointed out that he used to patrol that airspace with live weapons and didn't want to get shot down by his former colleagues so I gather they felt atc communication was a high rather than low priority.