### 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)

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from airflow.compat.functools import cached_property
from airflow.configuration import AIRFLOW_CONFIG, conf
from airflow.datasets import Dataset
from airflow.exceptions import AirflowException, ParamValidationError, RemovedInAirflow3Warning
from airflow.executors.executor_loader import ExecutorLoader
from airflow.jobs.job import Job
from airflow.jobs.scheduler_job_runner import SchedulerJobRunner
from airflow.jobs.triggerer_job_runner import TriggererJobRunner
from airflow.models import Connection, DagModel, DagTag, Log, SlaMiss, TaskFail, XCom, errors
from airflow.models.abstractoperator import AbstractOperator
from airflow.models.dag import DAG, get_dataset_triggered_next_run_info
from airflow.models.dagcode import DagCode
from airflow.models.dagrun import DagRun, DagRunType
from airflow.models.dataset import DagScheduleDatasetReference, DatasetDagRunQueue, DatasetEvent, DatasetModel
from airflow.models.mappedoperator import MappedOperator
from airflow.models.operator import Operator
from airflow.models.serialized_dag import SerializedDagModel
from airflow.models.taskinstance import TaskInstance, TaskInstanceNote
from airflow.providers_manager import ProvidersManager
from airflow.security import permissions
from airflow.ti_deps.dep_context import DepContext
from airflow.ti_deps.dependencies_deps import SCHEDULER_QUEUED_DEPS
from airflow.timetables._cron import CronMixin
from airflow.timetables.base import DataInterval, TimeRestriction
from airflow.utils import json as utils_json, timezone, yaml
from airflow.utils.airflow_flask_app import get_airflow_app
from airflow.utils.dag_edges import dag_edges
from airflow.utils.dates import infer_time_unit, scale_time_units
from airflow.utils.docs import get_doc_url_for_provider, get_docs_url