Universal background checks wouldn't have prevented a single shooting, all shooters obtained their guns legally so they went through the background checks and waiting periods.
If the background checks in question are already there, but even you have your guns anyway, then what is your problem with them? If they are not there yet, how were they supposed to stop the shooters? You're pretty much stating that a regulation doesn't work properly because it's too weak to have a real effect/not there at all and use it as an argument why it shouldn't be given more power/introduced.

It's like arguing that it's not worth to turn on a turned off light bulb because it's dark in the room, so the light bulb clearly isn't working.

An actually it’s more like 1.5 million lives saved annually through the defensive use of guns
[Citation needed]

Also, the vast majority of mentally ill people are non violent
Sure, but the risk of violence or suicide is still increased as compared to general population. Let's stick to mood disorders for now and see for example https://www.hhs.gov/answers/mental-health-and-substance-abuse/does-depression-increase-risk-of-suicide/index.html
it is estimated that about 60 percent of people who commit suicide have had a mood disorder
By doing a general screening to find relatively few people with mood disorders (let's say 10% of the general population) and then giving them more attention you can give more attention to about 60% of people who commit suicide. By screening more thoroughly 10% of people you can screen more thoroughly 60% of a group that is significantly more likely to harm themselves with a gun. To me it sounds like a good idea.