Is there any limit on how far this aspect of the program should go? Is total elimination the only outcome that will slake this thirst?

Gun control advocates aren't monolithic. I'm sure there's a vast and wide diversity of opinions on the subject, even within the gun control advocacy community. Not everyone thinks prohibition is either desirable or feasible.

Some folks might argue for a system like Germany's - where firearm ownership is pretty widespread, but you have to demonstrate that you meet a bunch of criteria in order to acquire firearms on a "may-issue" basis, can only get them for certain purposes (and not just because you want one or a generalized desire for self-defense), and are subject to some broad categories of prohibitions. Some folks might argue for a system like Spain (similar in framework to Germany, but administered more restrictively) or support a system like Norway's (again, similar to Germany but administered far more permissively).

No doubt there are some who want a system like Japan's - near-elimination of private civilian ownership of firearms of any sort. But I don't think that everyone who advocates for gun control wants that. I believe most advocates would be perfectly happy if the U.S. could adopt something similar to what one might find in continental Europe (recognizing there's a wide diversity of regulations there). Borrow the firearms regulations scheme from Denmark or Germany or Spain and adopt it here, and I think organized gun control advocacy all-but-disappears from the U.S.