Open source is nothing more than an attribute of collaboration. Some of us love to call out that a project is publicly licensed by adding open source into the project name. While that does highlight the public intent of the project, it can be a distraction from focusing on the project purpose.
Collaboration can achieved without public code. In fact, unless you are committed to the whole process of running a software pipeline, holding public meetings, and assuming the public’s project co-ownership, I say don’t bother. Without the burden of open source code, you will be able to manage those aspects of your project management without interference.
However, by embracing the best practices of project management and collaboration, you will find that you end up running an open source style project.
The point here is, decide on what style of project you want to run before you decide on if the code will be public. Then if open source code is compatable with your approach, adopt an open source license.