Its research shows rising house prices hitting all parts of the country, not just London and the south-east.
A new style of building, modelled on that of Victorian philanthropists, is urged, where short-term profits are replaced by long-term community gains.
Ministers agree the system is broken and want to make housing affordable.
The government is investing £50bn by 2021 in housing loans and subsidies, but more than half of this is being earmarked for "market-priced" housing.
To arrive at the figures on regional housing affordability, Shelter used regional data on gross household incomes, new-build house prices, information on loans and advances from the Council of Mortgage Lenders and ONS statistics.
It said the problem was worst in the West Midlands, where 93% of privately renting, working families could not afford to buy a newly built home, even if they used the government's Help to Buy scheme.
There the average price of a new home was £206,950.
Second worst was south-west England, where 89% of people were pushed out of home ownership by price, and third is the East Midlands, where 87% could not afford the average £215,000 price tag.
Shelter says the current system works in favour of housing developers and land dealers, who aim to maximise profits from house sales by maintaining the shortage of homes.
Newly built homes are the main way the government is increasing housing stock.
Under what it describes as a "speculative" model of house building, Shelter says housing developers are required to maximise returns on their substantial investments and therefore cannot risk lowering the prices of the homes they build for sale.
Therefore they only add to housing stock gradually - with the result that new-build homes are nearly a quarter more expensive than "second-hand homes", Shelter says.
This has left ministers "tinkering" at the edges of a broken system.
Shelter's interim chief executive, Graeme Brown, said: "Big developers and land traders are making millions from a rigged system while families struggle with huge renting costs and have to give up on owning a home of their own, which has become nothing more than a pipedream.
"For decades we've relied on this broken system and, despite the sweeteners offered to developers to build the homes we need, it simply hasn't worked."
The charity argues the way to fix the housing crisis is for the government to champion what it is calling a new "civic housebuilding" programme across the country.
This would mean giving cities and larger local authorities the right to create "new home zones", where land can be bought at the value of its existing use.
And public, private and voluntary organisations would be encouraged to work together to build homes for the public good, rather than private profit.
It cites examples of homebuilding by Victorian philanthropists like Sir Edward Guinness, who donated £200,000 to set up the Guinness Trust in London, and the Georgian new towns of Edinburgh and Bath, the Edwardian Garden Cities and the post-war "new towns" and garden suburbs.
A Department for Communities and Local Government spokesman said: "We've already got the country building again with the highest number of housing starts for nine years. And since 2010, government-backed schemes have helped more than 362,000 households to buy a home of their own.
"But there is more to do and our Housing White Paper will help do just that, making sure this is a country that works for everyone."
It added that its reforms would help build more well-designed and affordable homes, encourage smaller scale building and give local councils the powers they need to get homes in the places that people want to live.
A spokesman for the Home Builders Federation said the only way to address the housing crisis was to increase supply over a sustained period.
He said there had been a 52% increase in the number of homes over the past three years, but added that more needed to be done.
"Help to Buy has played a significant part in the increase in supply by helping over 100,000 people, the vast majority of who are first-time buyers, purchase their own home.
"The industry is committed to delivering further increases in supply and more of all types and sizes of homes such that we can better match supply to demand."
Nearly eight out of 10 families across England are unable to afford newly built homes in their local area, a report by housing charity Shelter says.