There are really only two possible answers to "how should we do things in our company?" The first is that the politics and governance of your business is an internal decision that will be unique to your organization, and can't possibly be answered by strangers on the Internet. The second is that, from a project management viewpoint, responsibilities and charters are defined by the organization during Project Initiation.
To move forward, it sounds like your project should review and update its governance documents. As part of that effort, it should also review the charter and chain of authority that underpins the steering committee.
Review and Update Governance Documents
Whether you follow PRINCE2 or another framework/methodology, all projects should have an initiation phase. This phase is generally concerned with chartering the project, including defining the project's scope, objectives, controls, and governance.
From the tone of your question, it appears that your organization has somehow arrived at a point where the lack of an effective Project Initiation Document (PID) or project charter has been revealed as a pain point. Alternatively, you may have such a charter, but have outgrown it without updating the governance structure to meet the new realities the organization is facing. Either way, the solution is largely the same.
To fix a project or program governance problem, leadership needs to collaborate with stakeholders to capture the current baseline (e.g. the as-is governance structure of the project) and then inspect-and-adapt to meet current requirements. Specifically, leadership should document elements like:
- The project's current objectives.
- Who is currently responsible for what.
- How the project teams are presently structured.
- What the project's controls and governance structure are measuring.
Then the stakeholders should do a comparison of the current state with the desired future state to identify gaps and variances. This analysis should then result in an updated project charter that addresses the gaps and lays a new foundation for governing the project.
Review the Charter and Chain of Authority for Steering Committees
Note that steering committees don't appear out of thin air. In general, a steering committee is either a standing committee or deputized for the life of a particular project. In either case, such committees operate on delegated authority that is authorized (directly or indirectly) by senior management. The who, how, and why of the steering committee is a business/political decision that should be clearly documented in the project's charter.
As part of your process update, make sure that your steering committee is properly authorized and chartered. Every project needs a sponsor, but steering committees are not de rigueur. If your organization wants one, then it should also ensure that the composition, roles, and responsibilities of the committee are clearly defined and that its operational authority is clearly traceable.