The US doesn't seem to have a problem identifying the mentally-ill. In fact, it seems we prefer to embellish the rate of psychological problems.
The real problem is that private/public psychologists refuse to register patients on NICS, which allows people like Jared Loughner, who failed a military psyche evaluation, to buy handguns and commit mass murder.
The vast majority of people who fail military psych evaluations do not commit murder. I'd also venture to guess the vast majority of people who do commit murder have never had a psych evaluation.
You can thank Reagan for the gutting of mental health facilities.
His policies probably contributed, but this is in large measure the result of policies ever since the 1960s when Medicaid at its inception severely limited coverage for psychiatric residential facilities.