### 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)

"""
    FAB does not know how to handle columns with leading underscores because
    they are not supported by WTForm. This hack will remove the leading
    '_' from the key to lookup the column names.
    """
    def __init__(self, obj, session: Session | None = None):
        super().__init__(obj, session=session)
        def clean_column_names():
            if self.list_properties:
                self.list_properties = {k.lstrip("_"): v for k, v in self.list_properties.items()}
            if self.list_columns:
                self.list_columns = {k.lstrip("_"): v for k, v in self.list_columns.items()}
        clean_column_names()
        
        for obj_attr, desc in self.obj.__mapper__.all_orm_descriptors.items():
            if not isinstance(desc, AssociationProxy):
                continue
            proxy_instance = getattr(self.obj, obj_attr)
            if hasattr(proxy_instance.remote_attr.prop, "columns"):
                self.list_columns[desc.value_attr] = proxy_instance.remote_attr.prop.columns[0]
                self.list_properties[desc.value_attr] = proxy_instance.remote_attr.prop
    def is_utcdatetime(self, col_name):