Thursday's meeting hinted it remained unlikely the council would move back to the site.
Councillor Graeme Taylor, who moved the motion to seek tenders, said the decision was about more than the seismic safety of the building.
"We should also be cognisant of the fact that, in doing this, in passing this recommendation, we can move forward to revitalise a part of the city and the CBD that over the past two years has been described to us as starting to become a ghost town."
The building, which was built in 1968, has been closed since 2017, when engineering group Strata deemed it to be earthquake prone.
The decision to tender the demolition was originally meant to be made on December 5, but was put off after Robinson told the council a 2010 report he had done found the buildings to be safe.
His report found the closed council chambers on Hastings St would have a life of about 20 years before they could potentially be earthquake prone (now 10 years), and an adjacent building on the site had an indefinite lifespan.
Councillor Api Tapine said Robinson had brought valid questions to the table.
"There were some concerns that I shared around."
However, he said after a further meeting with Strata, he felt confident he understood why the reports had different outcomes.
"I'm now sitting in a position where I have good confidence in both reports and understand the difference between the two."
Robinson said he was curious as to what had happened at a meeting on Wednesday between council, Strata and Aurecon (who undertook a peer review).
"I haven't seen the information, I haven't seen the outcome or any reports that were produced to the council."
He also questioned the $75m that council was estimating it could cost to refurbish the council buildings.
"There is a cost to re-locating council - if they are going to build a new standalone building, then I would be interested to know what the cost of that will be."
At Thursday's meeting, Taylor also said: "Although much of the discussion has been centred around seismic assessments, I think it is also very important to remember that this is not just about a seismic report.
"There was a very comprehensive document about site optimisation and a business case done."
He said the council now had the opportunity to move forward.