Can someone provide a use case for the $controller service in AngularJS?

You can create common functions which are to be executed on $scope into one controller may be named 'CommonCtrl'.

angular.module('app',[]).controller('CommonCtrl', ['$scope', function($scope){
      var self = this;
      $scope.stuff1 = function(){

      }

      $scope.stuff2 = function(){

      }
      self.doCommonStuff = function(){
               // common stuff here
               $scope.stuff1();
               $scope.stuff2();

      };
      return self;
}]);

And inject this controller in other controllers let say 'TestCtrl1' like

angular.module('app',[]).controller('TestCtrl1', ['$scope','$controller', function($scope, $controller){
        var commonCtrl = $controller('CommonCtrl',{$scope: $scope}); // passing current scope to commmon controller
        commonCtrl.doCommonStuff();
}]);

Here, the in second argument of $controller service, we are passing dependencies that are required by CommonCtrl. So the doCommonStuff method will use TestCtrl1 controller's scope.

To mention one, it is useful in creating the target controller during unit testing.

Lets say you have a controller with signature .controller('MainCtrl', function($scope, serviceA){..}).

In testing,

// ...

beforeEach(inject(function ($rootScope, $controller, serviceA) {

  // assign injected values to test module variables
  scope = $rootScope.$new();
  service = serviceA

  // create the controller, by passing test module variables values as dependencies
  $controller('MainCtrl', {'$scope': scope, 'serviceA': service});
}));

it('test on controller', function() {
  //...
});

For more info checkout: https://docs.angularjs.org/guide/unit-testing