### 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)

if dt is None:
                        break
                    if dt.year != year:
                        break
                    if dag.end_date and dt > dag.end_date:
                        break
                    dates[dt.date()] += 1
            else:
                prev_logical_date = DateTime.min
                while True:
                    curr_info = dag.timetable.next_dagrun_info(
                        last_automated_data_interval=last_automated_data_interval,
                        restriction=restriction,
                    )
                    if curr_info is None:
                        break  
                    if curr_info.logical_date <= prev_logical_date:
                        break  
                    if curr_info.logical_date.year != year:
                        break  
                    last_automated_data_interval = curr_info.data_interval
                    dates[curr_info.logical_date] += 1
                    prev_logical_date = curr_info.logical_date
            data_dag_states.extend(
                {"date": date.isoformat(), "state": "planned", "count": count}
                for (date, count) in dates.items()
            )
        now = DateTime.utcnow()
        data = {
            "dag_states": data_dag_states,