Analogy with this would be supermarket self-service tills that offer the "Do you want a receipt?" option.
I generally select 'No' unless I'm purchasing something of value (e.g. alcohol, razor blades, etc.). No supermarket security person is going to chase you out of the store to check you paid for your carton of apple juice!
That isn't an complete analgy although there are some similarities:
1) There's no written requirement that requires shoppers to carry a receipt when leaving the store, (and in recent times, some shops are actively discouraging shoppers to take one anyway).
2) The conditions of travel on the railway expressly require passengers to present a ticket or authority to travel when asked by an authorised member of staff. That is what passengers buy into when purchasing the ticket.
3) Supermarket security staff are there to prevent shoplifters exiting the store when staff have a suspicion of such activities. It used to be the task of shop employees/owners but in modern times, the element of safety has resulted in contracting 'specialists' for that rolse at least to assist.
4) In respect of 2) above, if a person has managed to avoid being asked to show a ticket (not necessarily illegal), but following observation, there is a suspicion that they don't have the necessary authority to travel, stopping them in the environs of the station that they have just exited would have the same justification as staff/security persons stopping a suspected shoplifter outside a shop, i.e both are part of a theft prevention activity. In a large station, it is possible that BTP might be available to add weight to what otherwise would rely on a citizen's arrest.
Aside from the above, apart from a lack of education or a desire to dispose of incriminating evidence, (i.e. the ticket wasn't authority for the full journey undertaken, as might be the case for 'doughnutting'), nobody here has offered any reason why carrying a small piece of paper/card even for a few minutes is so difficult. Arguments about (conflicting) notices on bins, the bins locations and made-up definitions of an RPO's jurisdiction might promote their social media status in their own minds, but wouldn't help much in an actual confrontation, - especially as regards to the' attitude test' which many insiders here say can go a long way towards an outcome.