I recently read somewhere that funeral photography is actually somewhat of a 'trend'. To me, that makes sense. Why would you only want pictures of the happy moments on your life? Fact is, most of us are just as much defined by the unhappy moments such as death, accidents or misfortune. It's a shame we don't have pictures of those moments, since overcoming them can be source of tremendous strength.
Personally, I would encourage people to take pictures at my funeral. It's the last mark we'll all leave on this world. Why not record it?
We also took pictures at an aunt's funeral a few years ago. Yes, that also included the open casket. We did it during a private moment before the funeral, so as not to disturb others who might find it... somewhat morbid. I don't have that particular hangup, as photographing dead people is a tradition as old as photography itself.