I am not sure the asparagus assassin would be big enough to get all the rhizomes. https://www.environment.nsw.gov.au/-/media/OEH/Corporate-Site/Documents/Animals-and-plants/Pests-and-weeds/asparagus-weeds-management-manual-130486.pdf does not include specific treatment for Asparagus officinalis but I have dug out a dense patch, careful to get all rhizomes, then followed up the next year to get seedlings. I have also used stem scrape and poison. However, it is spreading widely and can be hard to find and treat.
https://canberra.naturemapr.org/users/22862, I think the asparagus that is mentioned in the article to which you refer is A. aethiopicus (there is one CNM record by Mike); it probably spreads in the same way as the A. officinalis (birds for larger distances / rhizomes). Thanks Mike for treatment hints; I have a damaged back and 2 tennis elbows from parkcare work so I can't dig and need to use other means of control. What poison did you use?