(This is for my POV, I don't run calls due to age)
I agree, answer the question and move on. Flipping out on someone for not searching will just scare them off. (however I know it can be irritating at times)
I always have an "emergency" bag in my car with 50' of 8mm dynamic rope, carabeaners (for rope systems), a survival kit, trauma kit, general medical kit, etc
Sometimes I'll take my personal kit along too, depends on my mood. ANYWAY...
My trauma kit is small and just included gauze, abdominal pads, gloves, few band-aids, roller gauze, shears....I think that's about it.
However I also have other gear for non-trauma emergencies. Plus my person bag has a 'scope, BP cuff, more gauze, etc.