Animation would still play in the viewport.

There are two ways to unlink an action from the Outliner:
[1] Unlink Action on the Animation Data context menu.
This does outliner_do_data_operation / unlinkact_animdata_fn and has
the correct DEG update.
[2] Unlink on the Action context menu
This does outliner_do_libdata_operation / unlink_action_fn and was
missing the DEG update.

Now add the missing DEG update to the second case.