A lot of people don't like them, but I would look into these:
https://www.gardendesign.com/trees/arborvitae.html
I am thinking about putting some up behind our house. The key is they grow fast and fill in quickly. It's sad about those gum trees as they are great for shade, but I have head the same thing about the branches falling and causing damage.
Hmmm... I will look at those and see if there are some Australian equivalents?
There might be. I think there are a ton of different varieties. For me the important thing is that they grow fast and that they stay green through the winter. That's when the leaves fall off the trees and you can really see the other house through the woods.