Uh, going to have to explain the meanings of words to you.
Background check does not mean, "He filled out the form wrong"
What little background check they MIGHT have done is punching his name into a computer and seeing if his name came up as having any felonies, which he didn't.
That we rely on people to self-identify their addiction or mental health status is NOT a background check. It's actually, a flaw. Most crazy people don't think they are crazy, and most addicted people don't think they have a problem until they've gone into recovery.
Now, if you want a REAL background check on these things, hire an investigator to do piss tests, make people take psych evaluations, etc.
When I applied for my current job, they talked to a number of previous coworkers and employers, they did a drug test, they did a credit check.
When I applied for my mortgage, they inspected every financial document I had, including credit cards, previous mortgages, W-2, Tax returns and of course, a credit score evaluation through Experian.
That's a background check.