It's not surprising the nursing home didn't initiate CPR. It IS surprising that you somehow forgot ALL your equipment in the truck. Being an EMT 101 is being prepared to make patient contact, and you were not. Somehow, you approached someone you knew was in cardiac arrest without any of the tools that might reverse it. And, why did you think a backboard was required?
It's not your fault this person died, but you kinda just let them so you could be a racist to your student. How are you qualified to train anyone? I will say this is a situation that COULD have occurred, but only because all things are possible.