You have my sympathy. A coup is never pleasant and resentments can simmer for decades, but congratulations for being prepared to take this on rather than sitting back.
I am not familiar with the Victorian legislation but I expect it is similar to the ACT legislation. Here, the legislation just says “The committee must elect a chairperson.” without any further comment on process. I would say that if the committee has now elected you to be chairperson that automatically means that the person previously elected to be chairperson is no longer the chair.
I think you were right to offer the 7 days to provide evidence that you had not acted properly. Make sure the committee has minuted the decision to offer that with the date and at the next meeting minute that no evidence was presented. If you have the rest of the committee at least a solid majority behind you, you should be OK. Just be prepared for a lot of bluff and bluster from the deposed.
Does the managing agent have a boss or head office who could be phoned? Does the managing agent work out of an office somewhere? You could just show up and ask to see someone in charge if your manager does not appear.
What about the treasurer and secretary? Are they on side?
As for the matter of getting quotes etc. I don’t think it is necessarily wrong that your previous chair was getting quotes or researching particular matters. My experience on an executive committee has been that often it is quicker and easier for some matters to just organise some quotes myself on one matter while some other committee member might research and chase up quotes for some other matter. By the time you explain what it is about to the manager and they get someone to visit who needs to talk to someone on the ground anyway, often it would have been less work to have done it yourself. What is not good is if the chair was making unilateral decisions and not keeping the rest of the committee informed and involved.