Create menu link for authenticated user programatically
Given that you are simply moving a link of an existing route in a drupal menu, you are probably better off using the more simple hook_menu_links_discovered_alter(). A custom link/route is probably a bit of overkill.

But, presuming you do want to go the way you are in your example, you would need to have both a my_module.routing.yml and my_module.links.menu.yml

In the menu links file you might have something like:

my_module.hello_admin_user:
  title: 'Hello Admin User'
  description: 'See a Hello Admin User Message'
  parent: system.admin_config_development
  route_name: my_module.hello_user
  weight: 100


and in the routing file you might have:

my_module.hello_user:
  path: '/hello-user'
  defaults:
    _controller: '\Drupal\my_module\Controller\MyModuleController::helloUser'
    _title_callback: 'Hello Admin User'
  requirements:
    _permission: 'access content'
    _role: 'authenticated'
  options:
    _admin_route: TRUE


You've got a bit of a mix going in your example. Note that route names and links names need to be unique across the drupal system. Using your module name as the start is a convention but not required.