The Help page to read on this is Save a custom set of geoprocessing tools and I think the key sentence that answers your question is:
You cannot add system tools to a project toolbox; only custom script and model tools that you build can be added.
Alternative workflows are described there and summarized here:
You can save a custom set of geoprocessing tools within an ArcGIS project so the most important tasks of your project can be quickly and easily completed. The set of tools you save in a project can contain both ArcGIS tools and custom tools that you have built or that others have shared with you.
and
You can use a project template to import a set of tools from one project to your current project or to a new project, or if you want to always have the same custom sets of tools regardless of your project.