It's also worth noting the culture of Liverpool itself and the issues within the city over many years. I have relatives up there, and a couple of them did spend fairly long periods on the dole (6 months to 2 years - both are now in work). They have both said that although it wasn't explicitly stated, there was not really much attempt to force them into work by the job centre, because there wasn't a huge amount of work around, and generally a lot of the gaps in the labour market up there were for people with specific qualifications (e.g. radiographers at the hospital or engineering jobs at the port).
There is a complete contempt amongst many in Liverpool (which will include everyone from business owners to train guards, to the job centre staff) towards a lot of what comes out of Westminster / SW1, so the likes of Grayling (already a man considered a public joke) demanding DOO (irrespective of the issues around McGee / Zee) is going to be met with utter hostility, from far more people than just railway workers. Again this is historical, whether it's from the days of the Toxteth riots, to Hillsborough and so on.
Getting rid of railway staff (a stable and relatively well paid job for the area) will not go down well with anyone, even if the bean counters (who let's face it often have no idea about poverty) shout and scream.
Finally there is a common view in many parts of the country that "people should work for benefits" which could be translated into both a left wing and a right wing way. The logical conclusion to that is that "well it's better we pay people to staff things, even if we could get away with out doing so - because it's better to have people earning a wage and being productive and spending their money in the economy, rather than doing nothing". And of course this can be argued to reduce the effects on external services, whether it's mental or physical health, the police, housing departments or whatnot. If staffed stations reduce crime and anti-social behaviour then it's a win for everyone. If it also means that someone is doing a days work rather than sat on the dole or sat in the pub then it's also a win - both for them and society. And of course what's to stop people once they have valuable work experience on the station from applying for guard, driver, management and so on. There have been many people who started out on the platforms and ticket offices who have ended up with very senior positions in the railway, and used their front line experience to improve things once in senior management grades.
I have no idea if any studies of this ilk have been performed, but it would be interesting to know how much public money is saved more widely and more long term, even if there is fuss about "paying people for front line jobs we could technically do without". I'm certainly not going to bring up the private vs public arguments for railways, but fragmentation of the industry appears to have created far more jobs which if the railway was more unified, shouldn't need to exist - and many of them are far less useful to the farepayer than someone on a station.