### 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)

child_instances = [
                    item
                    for sublist in (child["instances"] for child in children if "instances" in child)
                    for item in sublist
                    if item and item["run_id"] == run_id
                ]
                children_start_dates = (item["start_date"] for item in child_instances)
                children_end_dates = (item["end_date"] for item in child_instances)
                children_states = {item["state"] for item in child_instances}
                
                
                
                mapped_states: MutableMapping[str, int] = collections.defaultdict(int)
                for mis in mapped_instances.values():
                    child_states = {mi.state for mi in mis}
                    state = next(s for s in wwwutils.priority if s in child_states)
                    value = state.value if state is not None else "no_status"
                    mapped_states[value] += 1
                group_state = next((state for state in wwwutils.priority if state in children_states), None)
                group_start_date = min(filter(None, children_start_dates), default=None)
                group_end_date = max(filter(None, children_end_dates), default=None)
                return {
                    "task_id": task_group.group_id,
                    "run_id": run_id,
                    "state": group_state,
                    "start_date": group_start_date,
                    "end_date": group_end_date,
                    "mapped_states": mapped_states,