When people used to speak of hunger – as an issue, not a sensation – they used to speak of faraway. Hunger meant the developing world, NGOs, imploring eyes on the TV news. It was, in modern times at least, a distant problem. So it comes as a shock that in the second decade of the 21st century we are speaking about hunger in Britain, right here right now.

Perhaps we were reassured by the notion that we live in the age of "relative poverty", when those who were officially deemed poor were not actually destitute but just had less than everyone else. The poor of our era were, we imagined, not Charles Dickens poor. They still had food on the table and a roof over their heads. Some of them even had the odd plasma TV, or so we read in the papers.

Perhaps that is why there has been something of a delayed reaction to the rise and rise of genuine hunger in this country. The unkind would call it denial. But it is becoming harder to deny.

This week the nation's most senior clerics told of what they are seeing every day, in the parishes where they and their colleagues live and work. Vincent Nichols, the newly elevated Catholic cardinal, branded the way the welfare system functions "a disgrace", while 27 Anglican bishops and 16 other Christian leaders blamed the government's benefits changes for a "national crisis" of hunger.

Predictably the coalition's defenders told the men of the cloth to back off, telling them they had no business poking their nose into such matters and should stick to "religion". Apparently they interpret the old Alastair Campbell dictum that politicians shouldn't do God to mean that God shouldn't do politics.

Perhaps they think churches exist to tidy up the hymn books and keep flowers in the vestry. In fact, the major faiths see their mission as nothing less than healing the world. So of course if they see people going hungry, they cry out. It is their duty.

It's their right too. Few institutions in our national life are doing more to deal with the return of a problem some might have thought we banished after the Depression, if not the Victorian age. Where do you think many of the more than 400 food banks run by the Trussell Trust operate? In church halls. I visited one this week, in Hackney in east London, no more than a three-minute walk from a pleasant green complete with upmarket cafes and a specialist Italian deli. It is a sight at once heartening and shaming. Heartening because it is good to see that, for all the talk of alienation and atomisation, for all David Cameron's one-time insistence that we live in "broken Britain", people are still willing to give up time, money and effort to care with great sensitivity and respect for those they have never met. Shaming because, as Cameron admitted when the subject was flood relief, "we are a wealthy country", and in a wealthy country people should not go hungry.

The pity of it is inescapable. I saw a man take home a couple of carrier bags filled with the most basic kind of basics: tinned sausages, a can of soup, a packet of pasta. All decent enough, each allocation calculated for balance by nutritionists.

Except you and I would not want to live on that for three days, which is how long it has to last. And the Trussell rules say that he can only get such help twice more. After that, he will be on his own.

The government's reply to the clerics is that they're on a moral mission to wean people off "welfare dependency" and that, in the prime minister's words, it "is wrong to penalise those who work hard and do the right thing while rewarding those who can work, but don't". It is worth unpacking each element of that.

Cameron's statement rests on the repeatedly implied assumption that the only people going hungry are those who have opted for idleness as a lifestyle choice, who could work but don't fancy it. This assumption is false. The majority of poor households include at least one person who works. As Rowan Williams, the former archbishop of Canterbury, put it this week: "People who are using food banks are not scroungers who are cynically trying to work the system. They are drawn from the 6 million working poor in this country, people who are struggling to make ends meet in low paid or bitty employment." Sure enough, the very first thing the "clients" I speak to at the Hackney food bank tell me – unprompted – is how desperate they are to work. One is an immigrant legally barred from working. The other is a former removal man now aged 60 who can't afford to buy the van he needs to get started again.

Yet the driving rhetoric behind the government's welfare policy focuses narrowly on those dishonestly claiming benefits, even though such fraud accounts for less than 1% of the money paid out. To punish that tiny, parasitical minority, hundreds of thousands of people – poor not because they're lazy but because of low wages, patchy, zero-hours jobs or rising food costs – are going hungry.

As for welfare dependency, on that logic any and all support is part of the problem: the minute you help someone, you risk making them dependent. The bishop of Bradford is surely right to argue that the government might as well be honest and "say ... we are prepared for people to starve and become destitute in order to achieve that longer-term goal" of ending welfare dependency. Like refusing to put out fires lest you encourage fire brigade dependency, it would at least have the merit of consistency.

The subtler argument from the right is the one that celebrates the food banks' admirable embodiment of communal self-help. Surely the church should be happy that neighbours, not some faceless state bureaucracy, are helping the hungry? Nice in theory, until you see the extreme pressure food banks are under, the days they run out of food, the fact that affluent areas often, and predictably, get more donations than the poor areas that need most. That's the trouble with a voluntary system, designed to give emergency help only: it cannot cope with what is a nationwide crisis.

The benign, big society view also fails to reckon with shame. I spoke to a Glasgow volunteer who told me she knows of men too proud to use their local food bank, who instead walk miles out of their way so they can get food for their families without being seen by their neighbours. They cannot afford the bus, and the walk home carrying bags of tins is hard, but it preserves at least some dignity. I learned too of the single mothers who fear visiting a food bank, thereby admitting they cannot feed their children, lest they be deemed incapable and their kids taken into care.

This was why Britons sought to put the Victorian era of charity behind us, why we decided that sometimes a state service is better: because there is less shame in claiming a nationally mandated benefit than in going to a church hall, being handed a food parcel and having to nod your head and say thank you.

Still, the shame is bearable if the alternative is you or your family going hungry. What has become of us, when that is the choice we offer our fellow citizens: dignity or food? And this in our wealthy, wealthy country.

Twitter: @Freedland