My grands are told, "No thank you, no more gifts." And I have done it. My 19 year old dgd does forget for awhile, but then it suddenly occurs to her that she forgot to say "Thank you" and calls me.
For parents of young ones, a gift that arrives in the mail requires a thank you for two reasons:
1) It's courteous.
2) It lets the sender know that the child got it.
I know most of you know that, but I know a few folks who came from what we now call dysfunctional families who, sadly, were never taught manners.