I'm not sure the environmental impact of fishing an invasive species to extinction is positive compared to the problems the species itself causes. Especially if you want that venture to be profitable.
By the definition of invasive species, eliminating them from areas where they were not native and are now harming the ecosystem doesn't require making them extinct.
Could always stop at the point you no longer consider them to be a problem...?
It's hard to hit the brakes on a moving train. Still potentially possible.
And yet we still use trains and train stations.
Point taken. Perhaps I should have referenced Pandora's box.