Summary

At this current point in time it is not possible to set a team-managed (formerly next-gen) access property (the one you can set in Project Settings > Details) via REST API, neither when creating the project nor afterwards.

There is currently no way to do a GET to pull the current access status, which is another common ask which the workaround below can not accomplish. 

Workaround

Is possible to use our internal API endpoint to set the project access of the managed project as advised in the following community post:

/rest/internal/simplified/1.0/projects/PROJECT-ID/summary

To change the access settings, you have to send a PUT request to the above rest API specifying the access level: "OPEN," "PRIVATE," or "LIMITED."