Looking at Verzion's FAQ about insurance (question #5) it says (my emphasis bolded):
Originally Posted by Hockey
If I file a claim through Total Equipment Coverage or Wireless Phone Protection, how quickly will I receive my replacement device?
For insurance claims received Monday through Thursday (for lost, stolen, or damaged devices under Total Equipment Coverage or Wireless Phone Protection) Asurion will ship your device for next business day delivery. Please click here for claim and shipping timelines.
Note: Next day/Saturday delivery may not be available in some locations.
For defective device claims, replacement devices may be obtained by visiting a Verizon Wireless Store or by calling (866) 406-5154.
I would consider what they are telling you to be a breach of contract. Maybe talking to the supervisor, getting their name, and then going to Verizon customer support would be helpful. I think they work very hard to resolve these issues. Otherwise you can threaten various other actions.
Btw, I had to laugh about how your phone got broken - a patient kicking you. Sorry, but I used to work as an RN in Critical Care and ED and your story bought back a lot of oddly fond memories. Glad you didn't get hurt.