Officially, it’s not a guilt tax. Westminster council prefers the term “community contribution” to describe the idea that its millionaire residents might like to make a voluntary donation on top of council tax. It is, they say, merely a chance for the wealthiest to “invest in their neighbourhood”. Perish the thought that they may have anything to feel guilty about.

But whatever you call it, attempting to appeal to the social consciences of the super-rich is surely a sign of changing times. That a flagship Tory council should be dabbling in new forms of redistribution is interesting in itself. That it began considering the idea a few months after the Grenfell Tower fire, which had some of Kensington’s more liberal-minded millionaires asking why their council hadn’t charged them more and housed their neighbours decently, is more interesting still, given that Westminster’s guilt money is earmarked partly for tackling homelessness.

And it was thrown into even sharper relief this week by the tragic death of a man who often slept rough in Westminster, inside the tube exit used by parliamentary passholders to reach the House of Commons.

His death provoked emotional reactions among MPs, who would have walked past him daily while he was alive. But once again it has the rest of us asking the Grenfell question: how can this happen in one of the richest neighbourhoods in Britain, on the doorstep of those ultimately responsible for solving the problem?

It has long been the nature of big cities that wealth and power rubs shoulders uncomfortably with poverty and helplessness; that on some London streets you can turn a corner and enter a different world entirely. But that uneasy equilibrium relies on the fortunate being willing to walk on by – literally, in the case of rough sleepers – telling themselves it’s sad but there is nothing they can really do.

The significance of the guilt tax is that, according to the council leader, Nickie Aiken, the idea came from wealthy residents themselves, who began asking last year if they could pay more. Most tellingly of all, she says it is most popular among those living in “the most expensive homes”, reversing the normal finding that tax rises are wildly popular only with people who won’t actually be paying them. This is starting to feel less like a conventional tax, and more like the biblical concept of guilt offerings: pay up, cleanse yourself of the perceived sin of unwittingly perpetuating gross wealth inequality, and perhaps you might avoid a plague of locusts.

The pragmatic attraction of a guilt tax is that it’s quick and achievable and it beats wringing hands

For judgment day seems to be getting closer. If even the rich now worry that they might not be contributing enough, then the case for what Ed Miliband called a “mansion tax” on millionaires’ homes becomes overwhelming. If politicians weren’t still so collectively traumatised by memories of the poll tax, such a change could then form part of a broader overhaul of local taxation. (If you’re wondering why Westminster didn’t just bump up council tax for those in the top band, by the way, the answer is: it couldn’t. Legally, council tax is meant to rise in all bands or none, and not unreasonably the council decided it would rather avoid increasing bills for the poorest families (and yes, even the richest London neighbourhoods still have pockets of poverty).

But while tax reform is crucial in the longer term, it is of no use to councils in crisis right now, thanks to the toxic combination of funding cuts and rising needs. When even Westminster, which for years prided itself on having one of the lowest council tax rates in the country, is forced to pass round the begging bowl, the game is up. Even Tory shires are creaking at the seams now, with Northamptonshire cancelling virtually all new spending in order to stave off bankruptcy, and Surrey facing a £105m funding gap.

Some Labour councillors, meanwhile, feel caught between the rock of austerity and the hard place of Labour activists ideologically opposed to the hard choices that may need to be considered in order to protect core services. Whether or not the answer is for Jeremy Corbyn to win an election, in practice that could be four years away – which means right now the choice is between sticking-plaster solutions and a gaping wound.

Relying on charitable donations, which could dry up overnight, to fund essential public services feels precarious and wrong. But the pragmatic attraction of a guilt tax is that, like the decision by the Manchester mayor, Andy Burnham, to donate part of his salary to a homelessness fund, it is quick and achievable, and it beats wringing hands.

It remains to be seen how many residents actually cough up. Nobody likes to be seen publicly spurning a good cause, but behind closed doors it’s sometimes a different story. Even if it does work in Westminster, it is unlikely to translate easily to neighbourhoods without large communities of morally troubled millionaires.

But this is a political litmus test all the same. It will provide a broader clue to whether the guilt and shock sweeping the country after Grenfell was a temporary reaction or a genuine watershed moment, one where the penny dropped that decent public services cost money and that a prosperity built on turning a blind eye to poverty can never be secure. For there’s only so long that the residents of places like Kensington and Westminster can carry on, to paraphrase Sir George Young, stepping over the bodies of homeless people as they leave the opera.

Those infamous words, uttered in 1991 when a previous Conservative government was under attack over a previous proliferation of rough sleepers around Westminster, have since become shorthand in the fuzzy collective memory for heartless elitism.

But the truth is more complicated. Young was no ignorant Marie Antoinette. He may have drawn an overly crude distinction between street begging and homelessness, but he was the housing minister who persuaded his then boss, Michael Heseltine, to increase hostel places and hire outreach workers instead of forcibly moving rough sleepers on.

However clumsily articulated, his guiding principle was almost the opposite of the one for which he is now remembered. It was that there comes a point in a civilised society when the evidence before one’s own eyes cannot be hustled out of sight. Wealth in itself is nothing to feel guilty about – but there is a price to be paid for letting it blind you.

• Gaby Hinsliff is a Guardian columnist