Just consider the parallels: you weren't the driver, have no first-hand knowledge of events and are taking your son's word...and the officer considering his reps had no first-hand knowledge of events and took the CEO's notes (a 'legal binding document' as they call it and signed by the CEO as being a true record) as being the truth.
In reality, what would you have expected the authority to do given that a copy of the BB wasn't submitted etc? But note the council's comment at the top of page 2 ..'if no further evidence is received'. They've left the door open to your son to provide this at the NTO stage, so let's take full advantage. (Have you actually posted all their response? IMO, the end of page 1 and the top of page 2 are disjointed?)
Based on your son's account, this is winnable, but he gives himself the best chance by setting out the full picture which we could then help him present in the most effective manner.
So, the full sequence pl from why his mother was being dropped off(when, why there and then, is this routine etc.), why he was escorting her(presumably not just a familial sense of responsibility but a necessity based upon her condition), where he escorted her, what did he do then(stay, return to car etc.) and the events at the car?
As alluded to by others, there is a statutory defence based upon 'assisted alighting' which has nothing to do with the display of a BB, so I suggest that if there's any doubt in his mind as to whether the BB was displayed(NB. the council's claim is that 'there was no physical badge displayed' and not that there was one present but obscured or displayed improperly), this is flagged up now.
And a copy of the PCN and original reps pl.