Labour would scrap free schools and bring academies under greater local democratic control as part of a plan to unwind Conservative education reforms that it says have created a legacy of “fragmentation and privatisation”.

The new policy will be unveiled by Angela Rayner at the Labour party conference on Monday, the first time that the shadow education secretary has presented her own structural reform plan in her two years in the job.

“The Tories’ academy system is simply not fit for purpose,” Rayner is expected to say. “Labour will end the forced conversion of local schools to academies, scrap the inefficient free school programme and instead focus on delivering what works to get the best results for pupils.”

Labour said that it would allow local authorities to build schools again and halt the free school programme, a flagship initiative of Michael Gove when he was education secretary under the coalition government.

Local authorities will be allowed to take control of failing academies or what it calls “zombie academies” without a sponsor if they wanted to. The party says that there are 124 academies which have been earmarked for transfer to another trust, usually because of performance concerns.

“The Tories’ fragmentation and school system has created zombie schools – caught between academy chains who are under no obligation to take them on and a government that simply washes its hands of the problem, refusing to step in and take responsibility,” Rayner will say.

Schools will no longer able to operate their own admission policies, ending what Labour says is a situation where local authorities are left with a duty to ensure that every school age child gets a school place yet are not always able to place children where they want.

In the longer term, Labour wants to bring all academies into a new system in which all schools will “follow the same rules … centered on giving schools, parents and communities an active role in decision-making”.

Labour sources that the exact long-term system of democratic control and oversight would be subject to a process of consultation that it would begin immediately. But insiders stressed it rested on the principle of the public control of education as part of a national education service.

Until now, Rayner has largely avoided the debate around structures. A year ago, she said she did not want to get “bogged down” on the merits of particular types of schools; she added that she was not “fixated on local authorities” as the solution.

The education spokesperson also called for social media companies to ban anonymous accounts, complaining at a fringe event organised by the Guardian in Liverpool that most of the people that abused her online did so without using their real names.

Rayner said that social media firms should take greater responsibility for their users and complained in particular that Facebook seemed to have indicated that politicians should accept a higher level of abuse.

When asked what she thought about social media, Rayner said: “One of the first things they should do is stop anonymous accounts. Most people who send me abuse me do so from anonymous accounts and wouldn’t dream of doing it in their own name.”

Rayner conceded that using real names would not stop abuse but “it would certainly help a little bit. I think they should do more, they do have a responsibility for online”.

The Labour MP, who was being interviewed by Guardian editor Katharine Viner, said that politicians were at the receiving end of too much online abuse and cited the example of a recent incident involving fellow MP Stephen Doughty last month.

“Stephen Doughty complained to Facebook about the abuse and threats he had received and they said they had a higher bar for politicians. It’s almost as if politicians should accept a certain amount of bullying and abuse online,” Rayner said.

Rayner, who is often talked about as a future party leader, was also asked about her political ambitions and whether she viewed herself as leadership material. She did not rule out a bid for the party’s top job: “Well I think anybody can achieve if given the opportunity to.”