My goldens (many years ago) would never greet anyone without a toy in their mouths, ditto my current rescue Mika (labx). My late Rhodie and current other rescue Chloe (shep/rottie/bc x?) had/have zero interest in greeting with something in their mouths. I guess it's a retriever thing.

My only suggestion about biting the wrist, shove a toy in his mouth. Some dogs just want to communicate with a "yum" in their mouths to show how happy they are!