No, you are not a monster. But it’s your sons responsibility to decide. You tell him that he was the beneficiary of the dogs love and loyalty and making that decision is the price that is paid. You can help him out, but it’s his responsibility to take care of this.
You need to rent the movie A Dogs Purpose. Tell your son being busy is no excuse. If the dog is going to be put down, it’s your sons arms that need to be around him. Not yours
He wants to be there, but he is having a problem doing it.
I think he is afraid that he will want them to stop.
and maybe not be able to go through with it.