Outgoing Tory leader Gordon Keymer called for councils to be given the right to revert to the committee system, in a debate on new local government structures at Tuesday's general assembly.
He said the government's imposition of the executive-scrutiny split, in the 2000 Local Government Act, had done nothing for services and led to a two-tier system of councillors.
He added: 'It's time to tear up the Local Government Act 2000 and allow local authorities to return to the committee system if they so wish.'
The motion, brought by Bournemouth BC, saying the act had undermined the status of councillors, was carried by a cross-party majority, though with many abstentions.
That was also the fate of a motion calling for the LGA to lobby government to work more closely with councils on asylum policy.
It had been brought by Fareham BC and Gosport BC, following a Home Office plan to establish a centre for asylum seekers at an old naval base across their boundary.
Both felt there had been insufficient consultation from government.
There was also support for a motion from North Wiltshire DC expressing concern over councils' ability to respond to affordable housing needs, following the abolition of Local Authority Social Housing Grant in April.
Funding will now be allocated by regional housing boards, leading to fears that local needs will go unmet and councils will not be able to plan effectively.
Richard Kemp, vice-chair of the LGA's housing executive, said in some regions, councils had three seats on the new housing boards, but in others they had none.
The LGA will be reviewing the new boards and the impact of the abolition of LASHG during the summer.