This is more of a "plausible scenario", but I think an important (unintended) downside may very well be singling out an already vulnerable community, people with mental health issues.
Governor Andrew Cuomo has summarised the SAFE Act as:
The SAFE Act stops criminals and the dangerously mentally ill from buying a gun by requiring universal background checks on gun purchases, increases penalties for people who use illegal guns, mandates life in prison without parole for anyone who murders a first responder, and imposes the toughest assault weapons ban in the country. For hunters, sportsmen, and law abiding gun owners, this new law preserves and protects your right to buy, sell, keep or use your guns.
Specifically for the mentally ill, the SAFE Act requires mental health professionals to report their patients to a local director:
What does the new MHL 9.46 require be reported?
MHL 9.46 requires mental health professionals to report to their local director of community services (“DCS”) or his/her designees when, in their reasonable professional judgment, one of their patients is “likely to engage in conduct that would result in serious harm to self or others.”
Who is required to report under MHL 9.46?
The reporting requirement extends to “mental health professionals,” defined in the law as four professions – physicians (including psychiatrists), psychologists, registered nurses, or licensed clinical social workers.
Who is potentially a subject of an MHL 9.46 report?
All persons receiving mental health treatment services from any of the four types of mental health professionals identified in the law, regardless of the setting in which they work, may be subjects of 9.46 reports.
While it's extremely hard to argue against the need to prevent the mentally ill from buying guns, the direct association of mental illness with gun violence is a dangerous one, and one that doesn't sit well with mental health professionals and academics:
"It has set back stigma a trillion years," says Sharon McCarthy, programme director for the Westchester branch of the National Alliance on Mental Illness.
"In developing this law, you brought in the mentally ill people," says Ms McCarthy, whose daughter is bipolar. "You didn't bring in the gangs. You pinpointed that group."
Like others in the field of mental health, Prof Applebaum1 fears that patients, who already do their utmost to maintain their anonymity, will recoil at the new regulations.
"Lots of people, because of the stigma associated with mental illness, don't want anybody to know that they're in treatment," he says.
"They don't use their insurance coverage. They pay out of pocket, so their employer and their insurer won't know. They may not even tell their spouse."
The thought that their doctor might now report them to a local official, he says, could be disastrous.
Source: Will gun laws hurt the mentally ill? - BBC
1 Professor Paul Applebaum is the director of law, ethics and psychiatry at Columbia University.