### Description

Going to Airflow web UI, Browse>DAG Run displays the list of runs, but there is no way to search or filter based on the text in the "Note" column.

### Use case/motivation

It is possible to do a free text search for the "Run Id" field. The Note field may contain pieces of information that may be relevant to find, or to filter on the basis of these notes.

### Related issues

Sorting by Note in List Dag Run fails: 
https://github.com/apache/airflow/issues/30041

### Are you willing to submit a PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

"This indicates that this deployment may be accessible to the public internet. "
                        "This warning can be disabled by setting webserver.warn_deployment_exposure=False in "
                        "airflow.cfg. Read more about web deployment security <a href="
                        f'"{get_docs_url("security/webserver.html")}">'
                        "here</a>"
                    ),
                    "warning",
                )
        return self.render_template(
            "airflow/dags.html",
            dags=dags,
            dashboard_alerts=dashboard_alerts,
            migration_moved_data_alerts=sorted(set(_iter_parsed_moved_data_table_names())),
            current_page=current_page,
            search_query=arg_search_query if arg_search_query else "",
            page_title=Markup(page_title) if page_title_has_markup else page_title,
            page_size=dags_per_page,
            num_of_pages=num_of_pages,
            num_dag_from=min(start + 1, num_of_all_dags),
            num_dag_to=min(end, num_of_all_dags),
            num_of_all_dags=num_of_all_dags,
            paging=wwwutils.generate_pages(
                current_page,
                num_of_pages,
                search=escape(arg_search_query) if arg_search_query else None,
                status=arg_status_filter if arg_status_filter else None,
                tags=arg_tags_filter if arg_tags_filter else None,
                sorting_key=arg_sorting_key if arg_sorting_key else None,
                sorting_direction=arg_sorting_direction if arg_sorting_direction else None,
            ),