### 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)

try_count = ti.prev_attempted_tries if ti.prev_attempted_tries != 0 else ti.try_number
            task_dict = alchemy_to_dict(ti) or {}
            task_dict["end_date"] = task_dict["end_date"] or timezone.utcnow()
            task_dict["extraLinks"] = dag.get_task(ti.task_id).extra_links
            task_dict["try_number"] = try_count
            task_dict["execution_date"] = dttm.isoformat()
            task_dict["run_id"] = dag_run_id
            tasks.append(task_dict)
        tf_count = 0
        try_count = 1
        prev_task_id = ""
        for failed_task_instance in ti_fails:
            if not dag.has_task(failed_task_instance.task_id):
                continue
            if tf_count != 0 and failed_task_instance.task_id == prev_task_id:
                try_count += 1
            else:
                try_count = 1
            prev_task_id = failed_task_instance.task_id
            tf_count += 1
            task = dag.get_task(failed_task_instance.task_id)
            task_dict = alchemy_to_dict(failed_task_instance) or {}
            end_date = task_dict["end_date"] or timezone.utcnow()
            task_dict["end_date"] = end_date
            task_dict["start_date"] = task_dict["start_date"] or end_date
            task_dict["state"] = State.FAILED
            task_dict["operator"] = task.operator_name
            task_dict["try_number"] = try_count
            task_dict["extraLinks"] = task.extra_links
            task_dict["execution_date"] = dttm.isoformat()