It is not unhealthy or insensitive (which is what "morbid" means) to know cause of death.
Knowing cause of death has value for public discussion about policy, health, and medicine. Even if it's something like suicide or AIDS, we'll know that we should be paying more attention to preventing those.
IMDB and Wikipedia always list cause of death when it's known. Are they morbid too?
The obituary writer is free to write it any way he wants, but you seem to be arguing that it's wrong to include the cause of death. The reason we're reading this particular obituary at all is because we want to know more about the person. How is a list of surviving relatives, his home town, or age any more relevant or important to mention than cause of death?