Writing mainly as an ex- teacher + work place first aid trained
With accidents, fortunately rare, the accident book was often filled in later - after dealing with the injured child. Accurately but later. It won't have detail of the actual injury - as they don't have x-ray equipment, medical knowledge etc.
They may need to ask others who were around as to what happened, what may have caused the accident, was it preventable etc.
From your description they did everything properly.
They may have downplayed matters when on the phone to you - not wanting to cause panic, rushed driving etc. His injuries may not have caused as much concern when they started the call.
Sometimes the pain from injuries develops as the adrenalin wears off/bruising develops. What seems initially just a graze type injury 'develops'.
This happened to me with a pupil. He'd fallen running for a bus. We spoke to him, looked at him. His only injury - from what we could see and what he said - was a grazed palm. It seemed best to get him on the bus and home soon - he wanted to. it looked a superficial injury - not really worthy of tan accident book entry.
As he sat down his knee started to bleed (no cut on trousers and no mention when we asked him) and as his adrenalin levels lessened his phobia of blood kicked in. (Secondary school - I and assisting colleague didn't teach him) 5 minutes into his journey he had screaming abdabs, bus took him to hospital etc etc. Head fully supportive of me and colleague.
Breaks are not always obvious - I fell and hurt my wrist last summer. Over the next week, I went rowing, did diy, moved stuff around, lived a normal life etc. On realising that it wasn't getting better took myself to A+E came home with a plaster.
When I started teaching in the early 80s workplace first aid covered a wide range of situations - childbirth, setting bones, bladed injuries, diagnosing strokes, poisonings. A couple of hours a week for a month. Each time I renewed my qualification something left the list. The last one I did was little more than CPR, ABC and how to call an ambulance.