For the most part, I believe your response is very thorough and reasoned. However, I would make 2 points:
1. Mental Illness, as you highlight, will be a very difficult thing to define and implement without triggering extensive numbers of lawsuits and another wave of gun purchases before the laws are changed. You not only need to get into HIPPA, confidentialities, and significant differences, but whatever outcome is generated will change behaviors. For example, what are you going to do on vets with PTSD ? (do you want people going to their doors are trying to forcibly take their guns?), Are you really going to stop people with bouts of depression from trying to protect their families? What about people who developed mental illness because THEY were abused in some way (raped, attacked, etc..) and they want and NEED protection to feel safe again? The phrase two wrongs don't make a right comes to mind. As for behaviors, . . your answers to these questions will influence future behavior, possibly negatively. If people know their hunting rights are going to be affected, they won't seek treatment when they need to, etc.. for fear of losing their guns. People with depression won't seek help to control it.
Many going through treatment were VICTIMS and will see this blanket decision as another ATTACK against them. (ie If you want to take EVERYONE's guns, so be it, but don't limit my ability to prevent a future attack against me) Bottom line: To those that have no mental illness in their family, I am sure this sounds like a very easy decision to make, but . . .it's not.
In theory, tracking guns sounds great, but do you really think those possessing guns will allow that to happen? No way ! They will hide them, bury them and DEFEND the taking of them. My fear is that too aggressive a stance will shift more people towards a radical/supremacist gun-toting leaning hell bent on depending their rights to the death - look what's happening already with Trump. (which is worse - going for overkill on guns and triggering a greater growth in these groups or preventing a reasonably small number of attacks? ) something to consider.
My guess is that the changes will have to be gradual. Start with those that have criminal records or those with severe mental illness (those who have been hospitalized, have attacked others, etc..)
The last issue with some of your suggestions is the RESOURCES needed to implement them - Similar to illegal immigration. Which is a higher priority - trying to take people's gun rights away that haven't violated any laws or pursuing those who have? (illegal immigration, violent offenders, those who are bigger threats) The answer sends a VERY powerful message to people, whether they want to pursue mental health help and can just drive one more wedge into the American fabric.