So the USAnians have loadza Gunz, because they like them, and the constitution says they can have them. But the problem is that nutters then shoot other people with those gunz. Another problem, more serious in terms of deaths but of course getting far less press attention, is that loadsa people top themselves using said gunz.
The constituion sayeth the familiar A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed. People don't like the unrestricted right given by the second half, and tend to argue that the first clause renders the rest void; but that doesn't seem to work. Also, people tend to like the idea of removing assault rifles out of the hands of people, even those are probably the most useful things for a militia.
So why not just take the first clause seriously? Allow people to keep their AR-15s, possibly even drop the restriction on fully-automatics; but in exchange for that ask for the "well regulated" piece to be taken seriously; which implies background checks and so on. CIP has some other ideas on what it might entail. I'm dubious that going all the way to you-must-join-the-national-militia would make any sense; indeed, that would effectively sabotage the proposal. My significantly weaker version would allow you to take gunz away from the mentally ill, permit background checks, allow you to oblige training courses, and so on. For all those reasons the NRA would oppose it, because they're the sort of people that want gunz-without-restrictions; but it would be a constitutional way to avoid unrestricted gunz.
[ps: I wrote this in February but have hastily finished it now as a response to CIP. Also by happy coincidence I now have a pic of our new weaponised cat available, and me wearing a tee-shirt proving my support for gunz]
* Armalite and ballot box strategy.