Many funeral homes have employees whose job it is to monitor the situation of immediate family members flying in for the funeral. As most airlines require proof from the funeral homes of the death of a particular person in order to discount fares, it is not unusual for funeral homes to be in contact with the airlines. If members of the family who are traveling, keep in touch with the funeral home, often it is possible to delay the service for a period of time to allow those persons to arrive.

Also, many funeral homes can assist people with getting discount airfares. Most people don't know that.

The misfire in this case is obvious Delta's, but also the funeral home should have been on the ball on this one as well.

I am very sorry for the OP's loss. Having a travel screw up adds insult to injury.