There is not a facility built in to remove the "People" link until logged in. You would need to edit the NAML code to achieve that.
As a stop gap, you can go to OPTIONS > USERS > CHANGE PERMISSIONS and remove the permission "Show_group_members" from any group. That stops members of the group being displayed, but works all the time even if logged in.
Volunteer Helper - but recommending that users move off the platform!
Once the admin for GregHelp now deleted.