As you said, it would be rather useless in some areas (most areas of Tasmania, for sure). It would also be useless in any area during windy weather (and in Tasmania when it rains, it's usually windy). So if you're going to take an umbrella, you'd need to take all the other wet weather gear as well.
I can see that it might be OK for sun shade on a day walk though.
I did see a guy using an umbrella on the way up Mt Ossa once (sunny day). I though he was a bit of a looney at the time.