In order to get a good job or rent an apartment, you need to supply references, other people who will vouch that you are a good person. So why not when buying firearms?
Government background checks only have information if the person has a documented history of mental issues. However, the people who know an unstable person most likely also know that they have issues, documented or not, and would deny them a reference.
Law abiding people who want to own a gun should have no problem getting references, and their access to firearms would not be affected.