Phil, my understanding is that Myxomatosis was deliberately introduced into Australia in an attempt to eradicate or control the European rabbit which was also introduced. Rabbits are a real problem here and Myxomatosis was a botched effort to fix the problem but it was not developed here. It is an awful disease and I have witnessed first hand what it does to the rabbits here on my own small acreage, of which there are quite a few I might add. It initially came from South America. There is now also RCD or calicivirus as well as myxo.
Introduced foxes are also a problem, along with feral cats and cane toads. Foxes and cats are doing unimaginable damage to the long term survival prospects of many small native species of marsupial like bilby, quoll, antechinus, possums and gliders with many now on the endangered list. The cane toad, introduced to control the cane beetle, has spread across Northern Australia and due to it's toxicity has caused a disastrous wave of death amongst the larger predators like snakes and monitors that try to eat it.
There are a several introduced species of plant that the early settlers brought over here as well which are a real biological / environmental disaster, those being Lantana, Privet and Blackberry and Camphor Laurel among others.
We are Rabies free which was the problem with Johnny Depp's dogs, although it is doubtful Pistol and Boo had rabies. There are a few very suspicious cases of Lyme disease in people who have not travelled abroad but the official line is that we don't have it here, which is probably wrong.
I am a strong supporter of strict biological border protection and quarantine, but immediate death of possibly endangered species, as you correctly point out is very unnecessary.