Tenant insurance underwritten with landlords name

[Kar Sun]

All my tenants are required to have tenant insurance. They have a liability for a full value of my property plus I make them have personal property for $30k. 
is it advisable to have the insurance underwritten with tenant and my name on it?

I believe if something happens and the tenant only has it in their name I am unable to get the proceeds because the check will be written to them. 
Additionally , the coverage should satisfy my mortgage company where I don’t have to buy a separate landlord policy. 
what say you?