Combine textual Button with rich Table

[heeplr]

I'm playing around with textual and try to implement a sortable rich Table widget where column headers can be clicked to sort by that column ascending/descending.

Since Column headers can be any renderable, my naive approach was:

from rich.console import RenderableType
from rich.table import Column, Table
from textual.app import App
from textual.widget import Widget
from textual.widgets import Button, ButtonPressed


class SortableTable(Widget):
    """rich table but with sortable columns"""
    table: Table = None

    def __init__(self, *headers, label=None, **kwargs):
        super().__init__(**kwargs)

        # list of columns with buttons to pass to Table() as headers
        button_columns: list[Column] = []
        # create buttons from headers
        for header in headers:
            # create button
            button = Button(label=header)
            # create column with button as renderable header
            column = Column(header=button)
            # append to list
            button_columns += [ column ]
        # create table
        self.table = Table(*button_columns, title=label, **kwargs)

    def handle_button_pressed(self, message: ButtonPressed) -> None:
        """A message sent by the button widget"""
        self.app.log(message)

    def render(self) -> RenderableType:
        return self.table

    def add_row(self, *args, **kwargs):
        return self.table.add_row(*args, **kwargs)


class MyApp(App):

    async def on_mount(self) -> None:
        table = SortableTable("col 1", "col 2", "col 3")

        table.add_row("foo", "bar", "baz")
        table.add_row("bla", "blubb", "blubber")

        await self.view.dock(table)

if __name__ == "__main__":
    MyApp.run()

This renders correctly but of course handle_button_pressed is never called upon click since the buttons were not mounted anywhere. on_click works as expected.

What would be the best practice way to have SortableTable receive ButtonPressed events?
Or am I doing it all wrong and there's a better approach to combine a rich element with a textual element in general?