The obvious question about the Garden Bridge is: where did it all go wrong?

The bridge, after all, should have been a lovely addition to the fabric of the city. An oasis of greenery in an area devoid of it, a new way of crossing the river and a new tourist attraction, akin to New York's High Line, all rolled into one. The Garden Bridge was not like the hilariously pointless “Emirates Air Line”, the cable car to nowhere which is even now ferrying empty pods between two windswept ex-industrial estates in a deserted bit of east London, like one of the follies listed by Marge Simpson at the end of Marge vs the Monorail. The Garden Bridge should have been great.

Yet in the years since it was first proposed, it's sunk further and further into controversy. The Garden Bridge Trust, the charity responsible for getting it built, has failed to raise enough money or acquire the land required to start construction before planning permission runs out this December. Official reports have repeatedly raised questions about the Trust's financial plans.

And today's news that London's mayor Sadiq Khan has written to the Garden Bridge Trust to tell it that the taxpayer would not provide the financial guarantees required for work to continue – effectively killing the scheme – is more likely to be celebrated than mourned. So how did something so lovely end up so loathed?

I will not be providing Mayoral guarantees for the Garden Bridge Project - my letter to the Chair of the Garden Bridge Trust. pic.twitter.com/VaK4t6yAe3 — Sadiq Khan (@SadiqKhan) April 28, 2017

The obvious explanation is the growing sense that the whole thing has been a bit of a con. When first the bridge was proposed, the intention was that it would be largely privately funded, with just a smidgen of Transport for London money required to get things moving.

The longer things went on, though, the more the ratio between those two sources of funding seemed to change. The predicted cost of the bridge continued to climb; yet the amount of money promised by private donors first flatlined, then began to slide.

So the amount of cash the taxpayer was going to have to put into this thing soared, with no end in sight: without a clear plan for funding the upkeep and maintenance of the bridge, it seemed likely to become a permanent line in the capital's own budget. As a result what had once been pitched as a gift to London began looking more and more like a pointless indulgence we would have to pay for ourselves. It felt like we’d been had.

But I think there's another, more philosophical reason why a lovely idea like a Garden Bridge should have become so unpopular: it fitted with a lingering sense that something has gone terribly wrong with this city.

We are, after all, in the middle of a housing crisis, which is seeing even relatively well-off people forced out of the city, and which has forced untold numbers to live in tiny under-regulated patches of squalor. The official definition of “Affordable Housing” has become a bad joke, yet new housing developments bend over backwards to avoid making even this limited provision. And in the midst of all this, the most visible property developments aren't much-needed homes for the masses, but commercial skyscrapers and luxury apartments.

Contemporary London prides itself on its tolerance and diversity and the way different social classes are all jumbled up together, without any of the ghettoisation seen in, say, Paris. Yet huge chunks of what look like public space are now private estates, often patrolled by private security. In our flattering, metropolitan liberal self-image, this isn't what London is meant to be.

It was, however, exactly what the Garden Bridge was going to be: a private garden masquerading as public space, yet funded by the taxpayer. The people most determined to see it built were a flotilla of rich, posh people: Boris Johnson, George Osborne, Thomas Heatherwick, Joanna Lumley. They were not us, but them – yet still they expected us to pay for it.

And then, once in a while, the bridge would close so that an investment bank or a private equity firm could throw a garden party, drinking champagne and eating canapes in full view of London as a whole, on a bridge we paid for but which we were not allowed to cross.

Perhaps the project isn't dead. Perhaps the Garden Bridge Trust will somehow find enough donors to get it finished without taxpayer support, and even find a way of funding its upkeep. Perhaps the rich really do want to give something back to London. If they do, I'm sure it'll be lovely.

But if they don't, I'm bloody glad we will no longer have to pay for it. This city has quite enough symbols of economic division as it is.