I suppose it depends what the purposes of justice are. The classical answer (OK, I checked online but it fits with my memory of lectures....) is
Retribution - Deterrence - Rehabilitation - Public Protection - Reparation.
In the case of the railway, I suggest that Deterrence and Rehabilitation (ie persuading the offender to mend their ways and making others aware of the risk is the railway's primary aim rather than Retribution, though Reparation has obviously become more prominent over the years.
So how do the 3 routes (warning via solicitor, settlement, conviction) pan out?
Retribution: someone who is let off with a warning has still paid a hefty price, as has someone who receives an out-of-court settlement through our good offices.
(the settlement will generally be lower than the solicitor's fee OR a conviction!)
Deterrence and Rehabilitation: are achievable to a similar degree via all 3 routes (I'm not suggesting 100% successfully, but given the number of unpaid fines we read about I wonder how effective a conviction is?)
:
Public Protection: not relevant here, unless it's the public purse being protected
Reparation: the OOC settlement benefits the railway more than a conviction, and using a solicitor obviously doesn't (!). (TfL take a different view from NR, which may reflect respective fare levels.)
Overall, the 3 different routes produce broadly similar effects, so I don't think the original premise is sound - no great wrong is done by employing a solicitor (nor by using our services!).