Millions of benefit claimants – who as a group fail to receive £5bn a year that they are due from the state – are being shortchanged by the welfare system rather than overindulged, a thinktank says on Sunday.

Rather than cutting benefits, ministers should seek to ensure that those on welfare receive their full entitlement, Demos says. Official figures show that one million people a year do not receive their full entitlement of housing benefit, equating to a failure by the state to pay out up to £3.1bn.

More than two million people a year do not apply for relief from paying their council tax bill, equivalent to more than £1.7bn in savings to the state. Meanwhile, the number of pensioners that were estimated to be entitled but not claiming pension credit was between 1.21 million and 1.58 million in 2009-10, worth a total of between £1.94bn to £2.8bn.

Those missing out on state assistance often end up as a burden to charities, the thinktank Demos says, and are not being properly helped into employment.

The report, The Ties That Bind, comes at a crucial time in the debate over Britain's welfare state. Last week it emerged that error and fraud in the benefits system is costing the government around £3.5bn. In response Lord Freud, the welfare minister, said that he expected there to be a record level of prosecutions in 2014.

But this week's report shows that the sum lost to error and fraud to those on benefits is dwarfed by the £5bn, at the most conservative estimate, owed but never paid out to those on welfare. On top of the benefits not claimed, statistics released last week show that the Department for Work and Pensions underpaid claimants, due to fraud or error, by around £1.6bn, which is a rise from the 2011 to 2012 underpayments of £1.3bn.

Demos says the government needs to change the law so that jobcentres should have a legal duty to ensure claimants are aware of their full benefit entitlement. The thinktank also recommends that the unemployed should be allowed to opt for help from recruitment consultants, rather than jobcentres, because of their poor record.

A customer service study by Consumer Focus found the jobcentre ranked second lowest on a public service satisfaction index – beneath the tax office and A&E departments.

Duncan O'Leary, the deputy director at Demos, who authored the report, said: "Our expectations of welfare have sunk far too low. Many people have paid into the system for years, yet are shortchanged when they need help. It should be the jobcentre's role to ensure people know what benefits they are entitled to. Different jobcentres could do this in different ways – from tasking benefits advisers with the job, to signposting people to other sources of advice. But they should be accountable for making sure people have access to the right advice.

"A reformed system should also give people more say over where they go for back-to-work support. At the moment people are stuck with the job centre for up to 12 months and then a work programme provider chosen by the government. This leaves people with no say over the service they are using. We no longer settle for that in other public services and we shouldn't do so with the welfare system."

A DWP spokesman said: "Every day Jobcentre Plus advisers up and down the country are successfully helping people to get the correct benefits and, crucially, to move off benefits and into a job if they are able.

"But today's welfare system is just far too complex with people having to make claims from councils, DWP and HMRC to get the right support.

"Universal Credit ensures more people receive the right support with a single benefit and overall three million people will be better off by an average of £174 a month."