Specify that when editing the template in the 3 Settings page.
This must be done when each new project is created from the template; the person creating the project should set Privacy to Public to team as shown:
Also, in the template > Project content > top right corner > click the images to edit Project members and ensure that the first “Members” line is set to Editor (or Project admin):
For more info on that, see also Introducing new access level in projects - “Editor”
Only with the API (or third-party tool, perhaps). But all projects in a team that has Privacy set as above are already taken care of. You can identify ones not set this way in project lists usually with a padlock icon and/or the word “Private.”
Larry