It's impossible to answer that question as it's asked

What you posed is way to ambiguous to seriously evaluate for any specific benefit or harm

Problems with "history":
a) The restrictions are going to be very porous. Mental disorders emerge more commonly among people under 30, often people under 25, but mental disorders can emerge at -ANY- time, including -after- a person has bought a gun. Mental illnesses can also resolve and the numbers suggest that survival to middle age is associated with decline in mental disorders. So status of history may not reflect the status of present risk very well.

b) Most people, somewhere around 75% to 80%, with mental disorders do not seek or receive clinical help, people who don't seek clinical help CANNOT HAVE A HISTORY. About 20% of Americans are estimated to suffer from symptoms of a mental disorder annually...that's 60 million people or 1 in 5, that's a huge number of people to treat as criminals in-waiting. That attitude would have social consequences.

Problems with "mental illness":
a) there are -many- mental disorders, most of them have no association with violence, in society or in private. The net that can be cast with this approach can easily be over-sized and complicate rather than resolve the problem of identifying the small number of persons who indeed are potentially dangerous.

b) not every mental disorder is yet known or accepted as an authentic disorder. The APA rejected inclusion of a number of mental disorders when creating it's new edition of it's diagnostic manual, including rejection of an anxiety disorder characterized by 'embitterment' which includes heightened likelihood of acts of deliberate vengefulness.

c) The likelihood of a person with a diagnosed -severe- mental illness (schizophrenia, bipolar, borderline, etc) committing an act of violence in the US is estimated to be slightly over 6%. The likelihood of social violence among the general population is about 5%. Which is to say even for persons with severe mental illness there is not statistically significant elevated risk. Indeed the possibility of random mentally disordered person committing an act of social violence is almost exactly identical to the possibility of any randomly chosen gun owner.

Problems with this approach.
a) This approach has much to do with exploiting existing stigma about mental illness in American culture and the need to create scapegoats (the NRAs 'monsters among us') so that cognitive dissonance (aka discomfort) about our society being violent can be reduced.

b) This approach actually increases stigma and fear about people with mental disorders which results

1) in people avoiding clinical assistance with mental disorders because they fear the harm done by such a label.
--this adds to the problem of untreated persons in society
--it confounds the possible effectiveness of this strategy by increasing the difficulty of finding people with histories of mental disorder

2) in increased discrimination against the mentally ill.

In case you didn't know it...2012 unemployment in the US among persons with mental illness averaged about 80%. Discrimination in employment reflects discrimination in association across society and results in marginalization and isolation of persons with mental disorders. This destroys a basic early warning system for potential problems with mentally disordered by reducing opportunities for interventions in gun violence by friends, coworkers etc. to near zero.

Increased stigma and discrimination simultaneously is quite likely to exacerbate feelings of frustration and unfair persecution that motivate even the mentally well to embittered acts of such as revenge.

Analysis of the effectiveness of the NICS database in preventing purchases from banned persons is that those on the list have one of the lowest rates of violation. Persons with histories of mental illness don't violate NICS. The real violations are among persons with criminal histories.

Suicide remains a dominant cause of gun deaths in the US. But it is not the chief concern or target of gun control which is driven by fear of gun violence committed on innocent others. But it is a serious problem that needs attention.

The mental disorder it is most often associated with depression. Depression is also the most common of diagnosed mental disorders in the US, and it accounts for the majority of diagnosed mental disorders per year...about 70% of cases of depression are diagnosed in women...who have lower rates of both social gun violence and suicide than men.

Because of the high prevalence of depression, over their lifespans well over half of Americans experience it personally. If these persons sought clinical assistance and thereby were placed on NICs, it would create an enormous burden on the reporting system would add to its expense and would provide little benefit other than the psychological relief that comes from having done 'something'.

Criminalizing depression would once again, contribute to increased stigma and people's perception that they need to hide or deny symptoms of mental disorder in order to prevent the social costs of such a diagnosis. The result would be decreased help seeking...the very things we might expect to help reduce the occurrence of suicide and the very thing needed to make denial of gun purchases to a person with mental disorder possible.

It is quite clear that some persons with mental disorders can be a danger to themselves or others. Protecting them from themselves, and protecting the public from them is an obvious and long-standing concern. Social systems already exist to respond to such people. But stigma and discrimination often drive these cases underground, leaving more general passive policies, such as waiting periods for taking possession of guns, one of the few possible responses. As we have seen in handfuls of recent mass murder cases, long-term planning that exceeds waiting periods is often involved as is coming into possession of a firearm that was purchased by someone else.

Our perceptions of the significance and fear of gun violence vary with our context. This week end just in the city of Chicago, over 6 people were killed in acts of social violence and 27 wounded by guns. In my rural county in SE WI there were no gun murders and no people wounded.

There are people with mental disorders in both places and in about the same proportion. Murders by guns have occurred in my rural county, and just as elsewhere gun suicides are many times more common.

Those patterns have actually been confirmed by research across America. The problem of social gun violence has causes that don't really center on mental illness, even while recent high profile cases of suicide plus mass murder do.

Impulsive acts of violence are part of human nature. Defense of territory and property is a part of innate human social behavior as well as learned behaviors even within the culture of the criminally inclined. Availability of guns provides options for tool use that has increased deadliness for those behaviors.

Identification of strategies that reduce social/criminal gun violence shouldn't be based merely on the size of the public's fear...which can be a massive illusion based on frequency or salaciousness in the media.

Strategies should be decided based on the greatest potential for reductions in social gun violence returned per unit community investment. That won't always mean that the biggest root problems will be addressed, but rather control of the biggest root causes that can be effectively implemented. Those strategies have to reach deep and engage human social behavior.

Focusing effort and expenditures for control of gun violence on broad swathes of the subpopulation of persons with mental disorders risks poor and ineffective use of resources. It creates attitudes that essentially criminalize illness. It often seems less aimed at reducing real social gun violence than it is aimed at calming a society that misunderstands mental disorder and has an irrational fear of it