In trauma (penetrating or blunt) the best assessment is MARCH

Massive Hemorrhage
Airway
Respirations
Circulation
Hypothermia

For medical emergencies it is still your ABC's.

With an Auto-Ped you are dealing with blunt force trauma and most injuries will be internal. You will not have much time with the patient to get down to figuring out what all is wrong with them, EMS rarely does as well. There is very little a bystander can do for blunt force trauma. Massive hemorrhage may not be seen quickly, it may be all internal. Ensure an open airway and adequate respirations best thing is keep the patient warm and treat for shock.

As as far as your reaction it is quite normal, the only reason EMS seems calm is it is their job, not their emergency. They are comfortable in those situations, but I promise you every EMT's first couple good trauma calls, they were reacting the same way you did.

Don't beat yourself up, it's difficult, do you best within your Knowledge, Skills, Abilities and Attitude and you will be fine.