Just nitpicking your particular example here, but rats actually did reach Australia on driftwood - the country has 7 species of native Rattus, as well as a variety of other endemic rodents. Rats are much more destructive on islands they didn't reach by natural means, such as New Zealand, while Australia's problems lie mostly with ungulates, rabbits, toads, cats and foxes. Humans were responsible for bringing them in, but upon their arrival they negatively altered the established native ecosystems - which, by definition, makes them invasive species. Yes, humans suck and introduced species aren't actively malicious (because no animal is), but this does not change the fact that they can be extremely destructive and are a problem that needs to be dealt with.
Regardless, this is an issue which people are always going to disagree on and I don't think it should derail this thread further.