I feel a little bit awkward about posting a comment as I've just been a lurker whilst waiting for my own case to go through. But whilst doing that (as something of a displacement activity, I suppose) I have tried to learn as much as I possibly can about Traffic Orders, the Tribunals etc, and felt I might be able to share my tuppence worth here.
Surely a key factor here is that the relevant articles of the Traffic Order "do not apply" where the "person in control of the vehicle" is "prevented from proceeding by circumstances outside their control".
This would seem to me to have not much to do with the Council's talk of an "exemption to leave an alleged broken-down vehicle unattended on a yellow line restricted area". The 'exemption', per the Traffic Order, would seem instead to hinge on whether the reason the vehicle was stopped was circumstances within, or outside of, the control of the person 'controlling' the vehicle. And, unless it is supported by precedent, I am unclear how the Council can claim that walking away from the vehicle during the period that recovery is being awaited kills that 'defence' and is not prima facie "allowed" (surely a matter of statutory interpretation, and not council policy). A reasonable interpretation of the Traffic Order to me (perhaps the Tribunal feels differently), would be that if a driver was reasonably awaiting recovery, there would be no reason under this Traffic Order they could not take a refreshment break, or even go inside their house, provided that the facts preventing their vehicle from proceeding had not changed in that time. Indeed that might be a sensible thing for a person legally in charge of a vehicle to do.
Unless there is tribunal or legal precedent on this matter, it seems to me that if the Council wanted to ensure that drivers remained with a broken down vehicle at all times, they should have sought to carve that out within the Traffic Order (if allowed under the enabling Acts)?
If the Council doubt the evidence and suspect that the receipt provided is irrelevant, or fraudulent, then presumably that is an argument they might develop at Tribunal stage and HC Andersen's suggestions, if followed, above would I am sure go some way to finding evidence that may disprove that. It's a shame there is no time on the recovery receipt. Regardless, (and again, it may be there is Tribunal precedent on this), I am unclear that the Google status of a particular recovery company is relevant here as opposed to the time that it was called, the time it arrived, and whether it seemed a reasonable option (e.g. calling a recovery company from Scotland might not be reasonable, but asking a friend to come from the next Borough with a tow-rope might be reasonable, especially given the response times of major recovery companies such as the AA/RAC etc).
The above is based on zero experience of Traffic Tribunals or parking cases, and there will undoubtedly be evidential and tactical things to consider, so I'll now pipe down and cede to the masters who know so much and help us for so little!