Warner and Smith each received one-year suspensions while Bancroft was banned for nine months following the ball-tampering scandal in March this year.
But an independent review commissioned by CA found the body to be "arrogant and controlling" and partly to blame.
In light of the findings, ACA boss Greg Dyer on Tuesday called for the banned trio to be given a reprieve, stating they had suffered enough.
Without Mr Peever's resignation sidelined players Steve Smith, Cameron Bancroft and David Warner have little chance of returning.
Longstaff's 147-page report, based on interviews with players, sponsors and other stakeholders, said Cricket Australia had consistently failed to live up to its values and principles and a culture of disrespect and bullying ran through the organisation.
The review was non-binding but he said he was considering the report's 42 recommendations, including an anti-harassment code to stop sledging and training to improve team leaders' "moral courage".
Particular focus is placed upon the fact that in neither the "How We Play" element of the most recent Australian Cricket Strategy, nor the former coach Darren Lehman " s blueprint "The Australian Way", is any reference made to the spirit of cricket or to respect for opponents and umpires.
CA is considering a recommendation urging CA-contracted players to be available for a minimum two Sheffield Shield and one grade match per season but the governing body rejected a bid to have players excused from Twenty20 internationals to play Shield and grade cricket.
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In Brazil's commercial capital of Sao Paulo, Bolsonaro's win was greeted with fireworks and the honking of auto horns. He also demanded that the rights of those who had not voted for Mr Bolsonaro be respected.
"There have been captains in the past who have been done for tampering with the ball and the penalties have been very lenient so there was no penalty for doing something wrong and it was always going to get to the case where it got out of control".
"You've got to be ahead of the game and know what the problems are and how they can be fixed".
Speed wants former national captain Mark Taylor to take over from Peever as CA chairman.
The players, who were involved in a bitter pay dispute with CA past year, felt they were treated as commodities, whose only value to the organisation was driving commercial success.
Smith, Warner and Bancroft have missed three Tests but the ACA's calls raise the possibility of them returning for the four-Test series against India, which begins in Adelaide on 6 December. "It's not our cricket team, it's Australia's cricket team, and I think for a little while, we lost that".
Katich told radio station SEN: "I think the players should see out their bans. We now need a strong board with a commitment to a way of being that is unimpeachable, that we can be proud of", while an unnamed CA staff member said "we are obsessed with being No.1 but it's fool's gold".
Dyer said that should convince decision makers to immediately lift the bans on the three stars.
Peever's refusal to publicly release the organisational review, chaired by The Ethics Centre's Dr Simon Longstaff, before he put his name forward for a second term as Cricket Australia chairman last week has been viewed around the globe as "arrogant" and underhanded.