I will list a few briefly off the top of my head, can go into more detail later.
Red flag laws that contain proper due process protections
Laws that share information among agencies so we reduce instances where department A knew one thing and department B knew something else but the dots never got connected.
CCWs that have reciprocity
A better background check system that doesn't have so many holes
1. Who pays for that? Right now, the FBI only has funding for the NICS system. States are not required to report into it, because funding is not provided.
2. Even those reporting agencies that are mandated (military, federal LE, ...) aren't willing/able to keep up with the reporting. Example: Air Force failed to report to FBI the mental state of the TX church shooter, and he was able to buy a gun.
3. States that do try to report to NCIS are 6-12 months behind at any given moment. That's a large window of opportunity to acquire a firearm.
4. Background checks only include information from agencies IF the subject interacted with them. If there was no arrest, no calls to authorities, no emails, no social media reporting, no mental health appointments, no drug or alcohol abuse treatment, etc., then the background check won't find anything. When the DoD does a Top Secret security background check, they send field agents to talk to references, associates and relatives you list on your application. They then ask those contacts for additional names of people that know you to be interviewed. Do you think that type of background check is appropriate for exercising a RIGHT? Might only take one person saying something questionable in an interview to wrongly deny you the right to own a gun. Secret clearances are based primarily on a National Agency Check (NAC) which only reviews agency records - no personal interviews with anyone other than the subject.
People who push for stricter background checks need to look into the last 20 mass shootings and all other gun-related violent crimes. Chances are, if they passed the NICS check, they'll pass any other background check you design. Those who fell through the cracks did so mainly due to gov't failures to follow existing processes. Creating more extensive systems and demanding more information won't fix the problem of gov't incompetence.
Assuming you can find 1,2 or even 3 people who passed the current background checks but would have been denied in a "better" system, at what cost? Funding, resources, and privacy are real concerns in any background check system, as is the potential increase in false positives (incorrect denials) that prevent applicants in real danger from obtaining the means of defending themselves.