It should be possible to set team-managed (formerly next-gen) project access (open, private or limited) via REST API

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