The correct argument is, if you can't prove that your restriction of a right would have a demonstrable and readily proven benefit, you shouldn't even argue for restriction of the right in the first place.
Now that is a "real" argument and a good one too.
The point is that it is NOT a restriction. There is no reason why sane, law abiding people in the country legally can't own a firearm. There is absolutely nothing to prevent it from happening by having a background check.