I am afraid our children don't get it because they are not the one in pain nor the one who has to make the decision to put your wonderful pet to sleep.
It is a very sad and brave thing for you to do. My daughter breeds show dogs, Doberman's and Boston Terriers. She understood and went with me along with my husband.
If your son does not understand just let it go as you are the one facing the loss of your good friend. Tell your son the decision is made and the Vet agrees with you.
Many warm Hugs for you.