I see some reasonable comments earlier, and I understand uncompromising attitude of the pro 2/a side .
So, what I think is that majority of mass shooters have either mental illness history, or a record of supremacist racist ideas posted on various social networks, third category of abusers are obviously various criminal profiles.
Simple psychological medical check that can be made in half an hour can vet out lunatics.
Good criminal record check, can vet the other two kinds.
If the line has to be drawn, it has to be done very carefully.
I will try to explain European system.
It is not good or perfect and is very restrictive - pain in the ass for legal gun owners, but in the same time it is not too bad. Explation follows.
AR15 as measuring stick, of the quality of the law:
Apart from USA, the question that I always ask myself, is the measuring stick of ar15.
If a country allows private ownership of ar15, then the laws of the country are at least acceptable in all reality.
(How many countries in the world allows ownership of ar15 or similar rifle? Not many)
Those countries are mainly: USA and most of EU countries. And few probably in South America, maybe even less in Africa. in Far east I am not aware of any country allowing ar15 type of guns.
So, this is maybe 10-15% of all the countries in the world
In most EU countries AR15 is legally obtainable.
Based on this, and since I am not yet moving to Arizona, Texas or Alaska (top states for gun owners as per guns and ammo magazine), i must say - most of EU gun laws or at least laws in my country I will say they acceptable trying to be realistic (conditional legal ownership of ar15). The worst countries for gun owners in Europe are probably Netherlands and UK (forget about ar15 there), best countries are now probably Austria and Czech Republic. Others are in between.
EU system and ar15:
So, European system has three levels to pass licensing, for all types of firearms
1. Reason to buy/own. 
This means one of the following reasons that has to be adequately proven:  1) collection, 2) hunting, 3) sport shooting and 4) not in all EU countries self defense, with very limited CCW options. 
Self defense is the hardest to prove, and if legally possible, this is mostly for active or retired LEO, active army personnel and vets, politicians, and persons in risky professions handling cash money and similar.
Generally speaking my opinion is that this part of licensing process can be removed. If person wants to buy any firearm for the reason of any kind, why would he not buy?
This issue of having a "reason" to buy, brings problem to heirs, who do not have sport shooting or hunting license and need to inherit the guns from diseased. Usually they cannot inherit if they dont produce some evidence of their gun-needing status. And they will have limited time to sort out their papers, or sell guns while they are in police custody. Police will hold their guns in custody for about 6 months.
usually they will loose, 6 months is too short.. 
2. Medical check.
Yes it can vet out lunatics.
But the problem is that medical requirements grow without reason. (I wonder who is making these???)
For example, diabetes, eye sight, hearing etc.... all together in my country total 18 diagnosis, without psychological diagnosis, as they are separate. (and to me all these 18 DG's they are irrelevant for the purpose intended)
So medical check can/should be restricted to psychological diagnosis, exclusively.
If medical check is not passed, then authorities confiscate guns, or not give permit to buy.
This means, if a person and elderly  gun owner has serious case of diabetes later in life, they will threat him as criminal and confiscate his property. (like diabetes is not problem enough?)
Each EU country handles details of confiscation differently. 
In my country, after guns are confiscated for this reason the owner has the right to sell them within 6 months, otherwise he can disable them or give them to government without compensation. During this period of 6 months, guns are in police custody
Selling large collection of guns in 6 months period is impossible, or high value guns to sell will be near to impossible to sell for true value.
3. Criminal background check.
Whatever they do, is reasonable, no issue there.
So, in my view removing the condition "reason to own" would not affect anything in safety.
But having medical requirements thick as medical encyclopedia is true risk for gun owners. We all get older, and none of us is getting healthier. So risk of loosing the guns and property is there with such regulations.
Having REASONABLE medical check, based on psychological evaluation + criminal background check could raise security a bit.
But- Just a bit.
it would never prevent criminals to obtain guns illegally on the black market, but it would at least took away anti gun argument for strict gun laws, when weapon abused was purchased legally.