Supervision can be a good idea to ensure sustainable design practices and materials are being used. Your builder is frequently called on to make decisions about materials and procedures that might vary from those nominated in the plans and specifications due to trade preferences or unavailability of preferred materials.
Builders refer these (often urgent) decisions to a supervisor for verification if one is nominated. If not, they may make expeditious but less sustainable choices.
Many designers offer a supervision service as part of their fee structure. However, project management or supervision adds substantial professional indemnity risk to a designer’s insurance profile, so many designers prefer a less formal advisory role.
Architects in some states are prevented from offering project management services. If you adopt this role as owner, consult professional advisers or consultants to verify your decisions.