### 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)

return redirect_or_json(origin, msg, status="error", status_code=400)
        latest_execution_date = dag.get_latest_execution_date()
        if not latest_execution_date:
            msg = f"Cannot mark tasks as {state}, seem that dag {dag_id} has never run"
            return redirect_or_json(origin, msg, status="error", status_code=400)
        tasks: list[Operator | tuple[Operator, int]] = []
        if group_id:
            task_group_dict = dag.task_group.get_task_group_dict()
            task_group = task_group_dict.get(group_id)
            if task_group is None:
                return redirect_or_json(
                    origin, msg=f"TaskGroup {group_id} could not be found", status="error", status_code=404
                )
            tasks = [task for task in task_group.iter_tasks()]
        elif task_id:
            try:
                task = dag.get_task(task_id)
            except airflow.exceptions.TaskNotFound:
                msg = f"Task {task_id} not found"
                return redirect_or_json(origin, msg, status="error", status_code=404)
            task.dag = dag
            if map_indexes is None:
                tasks = [task]
            else:
                tasks = [(task, map_index) for map_index in map_indexes]
        to_be_altered = set_state(
            tasks=tasks,
            run_id=dag_run_id,
            upstream=upstream,
            downstream=downstream,