There is no harm with you accompanying the supervisor when on site, but it is important to understand that only the supervisor, or somebody else they have delegated to (see clause 14.2), can decide whether there is a defect or not. And it is also important to understand that the term defect is a defined term (see clause 11.2(5)). You mention subjective judgements, but those should not be needed if the works information has been properly written. The test for a defect is not that the supervisor, or even you, does not approve of something the contractor has or has not done. The test of a defect is that the work is not in accordance with the works information or, in the case of works designed by the contractor, is not in accordance with the applicable law or the contractor’s design which the project manager has accepted.
Just because the supervisor or you would have preferred the works to be different does not mean there is a defect to correct if those works are in accordance with the works information. And, if the works information is silent about the works, for example not stating the use of a particular manufacturer’s goods, the fact they are not used will not be a defect. If the supervisor or you wants works that comply with the works information to be changed, the only recourse you have is to ask the project manager to issue an instruction to change the works information. For the example given above, that instruction would be to require that a particular manufacturer should be used. That will of course be a compensation event, and the contractor will be compensated for the cost and time it expends on this change.
All of this makes it clear that you and your designer must ensure that the works information in the contract correctly specifies and describes the works you require and states any constraints on how the contractor carries out the works (see clause 11.2(19)).