I'm a civilian and was the first car upon the scene of a tractor-trailer turnover three years ago. I pulled the driver out, and his wife was impaled by the guardrail below him.
Messed me up.
I'm on the board of the local fire district, and the fire station chaplain was a HUGE help to me. They can relate in a way that a civilian counselor probably can't.
Do talk to the counselor for the company.
What were the patient's chances before you arrived? Near zero.
What were the patient's chances when you got there? You pushed it to near 50-50. You did your best and there was nothing more you could do.
Thank you for your service to your community. While you are not on the ice and there are no pluses and minuses in your work, I do hope you get a save soon.
(But do process it before boxing it up and sending it away!!)
Emotions make us human, so they have to be dealt with...so deal with them, and get straight and square for that next human who desperately needs your skills and intellect.
Gallows humor seems brash to civilians, but it is what keeps many emergency responders and soldiers sane.
If it helps you, use it; especially if it helps you to voice your emotions, get them out, process them, and move forward.