I have been inside 2 big man made caves and there is air movement but clearly not enough to move such a big mechanism.
In fact one of those real life caves had to have extra ventilation because it was used to mine resources.
The idea of such a high roof cave when the cave supports are only the height of the door is ridiculous.
Not only do you need to explain the cave winds with magic troll farts you also need to use magic as explanation for the massive unsupported weight that is dug out around windmill.