Even with reuse it is pretty mating strategy dependent. It isn't a dial an extinction. Trying that with say feral cats would be a miserable failure or at best the equivalent of releasing a bunch of teaser tomcats into an area having only the TNR occupying effect on inhibiting population growth by claiming territory until they die of natural causes.
In short; a scientist demanding a cull will have more social impact than a politician demanding a cull. The more culls scientists demand, the easier it will be for fear-mongering politicians to demand culls.
Culling invasive species wherever they are invasive is something I support. But many people want to go a lot further than that.
As a species we’re piss poor at calculating blowback in complicated systems, ranging from the deaths of animal species, to interventions in foreign lands. Is it really surprising? We have an incomplete picture of the systems we intervene in, so the outcomes surprise us, then we rationalize our next blunder with the old saw, “this time it will be different.”