Are you sure they're not native? I was under the impression that they were in all of SE Australia except for Tasmania. It's pretty common on this side of the ditch and we've got a few on our land.
My experience with kanuka is that it's an opportunist and coloniser. Like its more famous cousin manuka, it takes over dry and scrubby locations and holds erodible soils together. As soon as something taller and more durable gets established and overtops it, it's pretty much gone. But we don't have fire as a part of our landscape except in rare circumstances here, so kanuka fills a niche and then gracefully gets out of the way.
If you really want/need it gone, I'd say cut it for fuelwood (it's one of the best out there) or biochar. Remember that much of the bush in your region is built to burn periodically and as a permie you find ways to manage around the characteristics of the place. Why use poison if a chainsaw will give you the same product?
My family has a place in Victoria that has lots of tea tree. They seem to have a fairly short lifespan, and over the past 20 years most of the trunks have collapsed and many of them have died. They don't seem to resprout readily when cut with the chainsaw either, so simply cutting them for firewood is definitely a viable option. I don't think poison need necessarily be a part of controlling these.
Regardig the native/non-native issue - my understanding is that they are native, but they were not nearly as widespread 100+ years ago. When the forests were cleared in the early colonial era the tea tree was a quick pioneer species of the disturbed land.
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