Well, depends on what these universal health checks resulted in. In an ideal world if it was found out you and/or your family need mental health help, you and/or your family would be enabled to get mental health help. And honestly yes, it is very likely you are not the best person to trust with a gun. Affect disorders are a red flag, half of personality disorders are red flags, disorders around psychosis are red flags. Just like eye problems are red flags for a driver's license.

If universal background checks would stop any of these shooters comes down to what those checks would really be. As far as I know on multiple occasions they had a fucked up life, were struggling and neglected for a long time. If you could work on un-fucking such persons' and their families' lifes while limiting their access to weapons, my bet is the situation would get better. Try not to give people reasons to go on a rampage in the first place.

But, on the other hand, I sort of see how this could get totally fucked up in America.

And there are more aspects, like it would help if you stopped making those people famous, repeating their names, showing their faces, giving the spotlight to the drama. Media should know better than letting future copycats know that they will get their five minutes if they kill people, but unfortunately they don't.