Show manage models action in the search input (#300165)

* Add filter actions to ActionList and manage models option in chat model picker

* Refactor model picker to support additional entries and improve manage models action handling

* Refactor model picker to conditionally add separator for additional entries and streamline manage models action handling

* Improve manage models action visibility logic in ModelPickerWidget

* Refactor model picker to streamline additional entry handling and improve action management

* Refactor model picker to integrate manage models action and streamline additional entry handling