I'm also very dubious of the ability of any sort of removal effort to make a serious impact on adaptable and fecund species such as carp.
Look at the long and only partially successful efforts to remove feral hogs and goats from various islands and parks. These are limited populations of large animals with low reproductive rates in isolated areas- about the best conditions you could hope for in terms of ease of removal- and these animals still often defy eradication efforts.
Now consider carp: enormous populations of animals with incredible reproductive potential widespread in vast interconnected drainage systems. Trying to eradicate carp from the American landscape would be like trying to eradicate dandelions, Norway rats, starlings, Chinese cockroaches, or any of the other high-r-strategy exotic "weeds". Not to say that there isn't a way, but trapping carp is like pulling up dandelions. Other methods must be sought.
There are certainly instances of successful elimination or near-elimination of once absurdly populous species. Passenger pigeons, bison, and cod are three familiar North American examples. We did it by destroying crucial habitat- feeding grounds, nesting areas, migration corridors- or by interrupting the food chain. The wholesale slaughter usually blamed for their declines is a secondary issue- salt in the wound.
However, these were species of established communities; carp, like other weed species, specialize in taking advantage of disturbed communities. Attempting to restore these areas to something like an undisturbed state might help, but unfortunately carp are very good at maintaining their habitat in a disturbed state. They uproot stabilizing vegetation and keep water turbid with their benthic foraging.
I mostly agree with you, and was just throwing that tradeoff out there for discussion.
In MD, nutria (a highly fecund, in mammalian terms, rodent) were abundant and widespread in Blackwater National Wildlife Refuge, and yet managers were able to implement a successful eradication program. How? Turns out they aggregate in the winter months to stay warms, and could be killed by the dozens. Although the Mississippi drainage seems impossible huge, this story offers a glimmer of hope.