I don't object to the idea of checking tickets at stations nor issuing penalty fares, except that is not how the role was sold to passengers. I am not sure that issuing penalty fares features as an aspect of customer service passengers want, especially when on some lines there is a belief you can buy tickets on the train. Well its not a belief, most of the time its a fact!
There does appear to be a need to do more ticket checking as that has in my experience reduced since the dispute started.
It is no secret that many Southern stations were (fairly) recently added to the PF scheme, and in reaction to a theoretical enhancement in said scheme, all stations gained a TVM at the very least, to allow passengers to prove intent to buy the right ticket before boarding; though stations such as those previously without any ticketing facilities are often those where it is still possible to buy onboard without hindrance anyway, and also those where TVM faults are resolved less frequently.
This has led to an unclear situation for passengers from a "payment culture" point of view, added to the fact that even the trained ex-revenue grade OBSs generally haven't issued PFs since early winter, and the fact that no change in policy on how to treat those buying on board seems to have been distributed to those retained as conductors.
Passenger Hosts, ie. rebadged revenue staff employed on the GTR routes under the TL & GN branding, occasionally make an appearance on roving patrols which take them around the Southern network, and definitely do still issue Penalty Fares, undertake MG11s, etc. This is a bit of a culture shock to passengers, though not unwarranted in some situations.
From a customer service point of view, it's my view that an enhanced policy is most certainly needed for revenue protection, and this may also pay back a return of reduced levels of antisocial behaviour, which improves the environment for customers as much as anything! I believe I once mentioned my observations on here that the difference in a regular and irregular staff presence was best seen by comparing late night and overnight services run by either Southern or Thameslink (formerly FCC, of course) on the BML corridor, and the improved environment and security visibly provided by a dedicated member of onboard staff such as a conductor was a great advertisement for them. This sort of reason is one of the pointers towards (at the very least) rostered onboard staff of some description.
Issuing penalty fares at stations could theoretically allow greater familiarity of OBSs with dealing with ticketless passengers, with whom some may well have felt uncomfortable until now, particularly new entrants - or ex-revenue staff not used to sticking to a dedicated train, or lone working! This could then be extended to their diagrams on trains, with guidelines on "when to stop" if they'd be otherwise unable to work the train safely. I'm not convinced it would be easy to devise such a solution, particularly given how closely-packed the station calls and passenger requirements are on some of the trains OBSs do work on.
There are signs that the revenue culture on the Southern network may change to include greater deterrents. There are mutterings about conductors being able to issue PFs on Envoy at some undefined future point, with OBSs presumably being able to do them electronically before then, and of course GTR's sister company Southeastern has recently switched to a policy of only selling full-price Anytime fares on the train, which is very slightly impacting Redhill to Tonbridge Line services.
All this being said - and I wish to use my words carefully - I can see that it would require great talent to reintroduce robust deterrents to passengers boarding trains without tickets, and sadly I suppose it may be seen that the GTR franchise may not readily attract that sort of skill! I think passengers should be ready for more severe penalties for not actively buying a ticket at the first opportunity, though. Ticketless travel from these people may become more common as trains get longer and busier, so those who thought they could buy on board don't end up doing so, and therefore work out that they'll probably get away without paying anyway. There's also every chance that a lack of enforcement could attract a general culture of lawlessness, which is most unwelcome!