As you say, we don't have the necessary detail, but my suspicion is that the OP's Mum and the station staff didn't reach a common understanding of what was supposed to happen (I'm not ascribing blame to either side). To the guard on the second train - who had no way to know what conversation had been had previously - the situation was simple: the passengers were on his train without a valid ticket.
Is it the contention that they should have been allowed to travel without a ticket because 'the man on the platform' said they could?
No. They should have been allowed travel because their booked train had been cancelled.
In that case it is far from unreasonable to suspect that some staff member may have allowed them travel part of the journey either to get them out of his/her hair or as a genuine attempt to help along someone inconvenienced by the initial cancellation.
The excuse of "the man on the platform said it's OK" should definitely not be allowed for people whose booked train runs but when it is cancelled it is obvious that booked passengers will have to find alternative trains to get to their destination.
I don't think that you and many others here appreciate how little many passengers understand or care about the minutiae of rail companies and ticketing nuances. The quote from the OP where the passenger's response to who they travelled with was "The Trainline" says it all. People like that look up their journey, buy a ticket (usually the cheapest one the can find) and travel as they are advised by the printout from the website.
If the original train hadn't been cancelled then they most likely would have travelled exactly as stated. It is only when things go wrong are they likely to fall foul of the overly complex ticketing rules that we all find so fascinating but if forced to deal with would have many occasional rail passengers reaching for the car keys.
The whole idea of barring passengers from subsequent services when their original train has been cancelled is just ridiculous, it is further penalising passengers whose plans have already been disrupted by an earlier cancellation.
I can see where a passenger gains an advantage from a cancellation say someone booked on a slow service switching to a faster TOC and arriving ahead of their booked time but other than that it is just being punative for no good reason.
If the TOCs are that keen to gain revenue or prevent other TOC passengers having a free ride on their trains then they could always instigate a procedure to retain or record all other TOC advances used following a disruption and bill that TOC directly.
--- old post above --- --- new post below ---
Whilst I don't disagree that advance fares would exist now without privatisation, I suspect that the current low level of many such fares is driven by the existence of competition between TOCs. This is the upside of competition which everybody likes, whereas the downside is that the tickets are more restrictive - in BR days such tickets generally had a geographical route restriction but now it is usually operator restricted, even after a cancellation.
While there are some flows that are probably cheaper due to inter-TOC competition these are far from the only ones offering cheap advances and would be only a very small % of the overall market.
The flipside of course is that we wouldn't have the price-gouging TOCs either (XC comes to mind) who use their position to price their flows much higher than average.