Thanks for those details @SJR – that really helps me zero in on what’s going on!
Because Local is a tool for quickly creating sandbox environments to work with WordPress sites, that “plus” button is actually just creating a new, plain WordPress site. This new site is using the latest version of WordPress, along with the default theme and the user that was specified during the create site process.
In order to import an existing site, you’ll need to somehow download a backup of the remote site and import it into Local.
For sites that are hosted at WP Engine or Flywheel, you can use Connect to make that process of pulling a site down easy. For other hosting providers, you’ll have to create a backup somehow using a plugin, or maybe tool that the provider offers.
From there, Local is pretty good about knowing what to do. Most of the time it’s as easy as dragging and dropping the zip backup onto Local which will prompt you for details of the environment.
For more info on the general process of importing a site into Local, see: