The QS is involved throughout the project from inception (when it is a glimmer of hope in the owners eye) to completion & closeout and even into operations, including forecasting, risk, and value. The project managers responsibility is the delivery of the end product. Project controls falls in the middle, sometimes overlapping with the QS role.
Return on investment, due diligence, risk, time, timing, quality, stakeholder expectations, planning, sustainability, contracting strategies, vendor selection, resource allocation, supply chain management, asset management, reserves and contingency, field administration, performance, change, claims, contract administration (I could go on for many paragraphs)...all contribute to project and lifecycle costs, and then there is the discussion about value....
Cost is everything, and everything is cost; "cost" cannot be marginalized as just one aspect of a project or asset.