If you don’t want to hand over control to a compulsory appointed strata manager you may consider applying for an adjudicators order against the owners corporation to carry out their duties under section 62 of the Strata Schemes Management Act 1996 (SSMA) which says;
62 What are the duties of an owners corporation to maintain and repair property?
(1) An owners corporation must properly maintain and keep in a state of good and serviceable repair the common property and any personal property vested in the owners corporation.
If the OC resolved at the AGM to obtain an expert report then the EC cannot over rule this decision. If the $5000 was only a budget allocation then that is not a sufficient resolution with respect of this and the committee does have the power to make this decision.
Getting back to your original question though, Schedule 3, Clause 4 of the SSMA sets out the process to sack one or more EC members. This can be done by special resolution at an extraordinary general meeting (EGM). An EGM must be convened ASAP upon receipt of a written request from at least one quarter of the owners (by unit entitlement).
Good luck.