Some things cannot be effectively regulated in this manner. At all.
There is simply no way to stop people from building their own 3D printers. There are too many open source designs, and they can be built with very simple parts that are readily available at the hardware store. Most hobbyist-level 3D printers basically come as a kit that they have to assemble themselves anyways. What happens next? Background checks to buy stepper motors? Background checks to buy a microcontroller?
To me this is like trying to mandate government backdoors in encryption algorithms. There is literally nothing that would stop criminals from just using an open source encryption algorithm that doesn’t have a backdoor, so you end up just making it so all legitimate communications are less secure than they should be.
Food, especially fresh food, used to be a lot more expensive when adjusting for inflation. A canned chicken like this doesn’t look super appetizing right out of the can, but it probably tasted OK after you shredded it and put it in a casserole. And it was significantly cheaper than buying a fresh whole roasted chicken, assuming you lived somewhere that fresh whole roasted chickens were even readily available. Food like this became particularly popular during the great depression, and stuck around for decades afterwards.
Nowadays, between industrialized farming, highly optimized supply chains, and a buttload of government subsidy, fresh food is comparatively cheap. You can get a whole roasted chicken right off the spit for $5-10 at just about any grocery store. So for most people the value proposition of a $3 canned chicken isn’t really there anymore, especially if you don’t have an enormous baby-boom-era sized family to feed.