The expensive problem identified at the start of the thread can be fixed as described (awkward). To clarify: The actuator (white gear moves) as temp is changed but it is no longer connected to the valve inside the tube. You can't diagnose this until the actuator is removed. If you can see the gear move but get no hot air it is likely the fault but removal confirms this. It is held by 2 5.5mm bolt head screws that are not easily reached.
Hot air can be achieved by manually turning the valve and leaving it open. You then have a car stuck on hot which is slightly preferable to stuck on cold.
I've chosen to not repair or bodge as a work around listed here gives me hot air. In my case and as described by others (so worth a try) some functionally can be achieved with the broken valve. To get hot: set climate to hi, face and feet. Turn fan on full and press recirculate. This pushed the valve open. You can hear it flop open. If the fan speed is reduced to 4 cold air returns. I can live with this.
If you want full functionally there is sufficient detail on how to reconnect the actuator on previous pages.