This could only be accomplished with universal registration. There are too many guns in the hands of the people for it to be any other way.
It would also essentially provide a slow roll for complete denial of any transfer of firearm ownership. On the one hand, we have a discussion about universal background checks. On the other, there are also discussions taking place about "deeper" or "more stringent" background checks. Both are very treacherous.
If "universal" background checks are the subject, why bother changing the law? If you are worried about the buyer's mental state in a private person to person sale, why not simply make the choice to go through an FFL voluntarily? Why must this behavior require legislation to be performed? In cases like Adam Lanza, no background check of any kind would have kept him out of his mom's safe. To that end, a deeper background check would not have prevented the sale of a gun to her unless probing questions were asked about her family, with judgments made about the state of mental health of others not involved in the transaction. IOW, her right to own a gun would arguably been rejected by 4473 approvers if they had known about her kid. She is denied her individual right because of some outside characteristic. That's a sickening application of government nanny-statism, and results in a denial of access to the second amendment.
If you are talking about the deeper background check, I am immediately suspicious of what that would entail. Absolutely no one has detailed what it would potentially mean to make a "deeper" background check. IMHO, engaging in and allowing this to start without knowing what the rules are is tantamount to simply handing over your right to own a gun. You immediately run the risk of moving from the equivalent of "shall issue" to "may not ever issue" if anything looks "not quite right". Who gets to determine that?
What does "strengthening" mean? What questions get added to the current 4473 list that we already have to fill out? How deep is the rabbit hole? How deep do the questions go? Who gets to make the evaluation and decision to approve the transaction, and what requirements would need to be provided to obtain the approval? Documentation of some kind? Show your papers, please! Will there be a grievance process for those wrongfully accused of not being fit to own a gun? What about those who have potential flags that look like flags but are really not? What about those who are perfectly law abiding but who have bad credit? Can they own a gun? Bad credit is a risk indicator. To some, it means a person cannot be trusted to be responsible with money, and that judgement is often applied to other areas. Should that be a determining factor in a person's desire to own a gun?
Here is the point. No one has articulated any detail pertinent to how these background checks would be applied and under what criteria approvals would be given. If that can't be done, I don't want to have anything to do with anything other than the law as it now exists.