It seems to me that anyone who believes universal background checks are a good idea must think that would be an effective way to prevent any criminal from ever obtaining a firearm. I don't think I need to list the dozens of reasons why that is one of the biggest myths of the anti gun ideology.
One of the comments on a story about Chicago said that every firearm used by a criminal originated with a legal sale, but doesn't get into how they ended up in the hands of criminals. I think they were suggesting that many legal buyers give or sell to criminals conveniently ignoring things like theft, which could include from the store before any legal sale has even occurred. I suspect that some folks think that if only we removed all guns from law abiding citizens then there wouldn't be any for the criminals to get ahold of.