As pointed out, a "mental health check" is not just a slippery slope, its the Highway to HELL.
Not much more than 50 years ago, homosexuality was a recognised mental illness. It was listed in the medical texts as such. While it seems implausible, it is not impossible that should enough like minded doctors and "mental health professionals" get together, gun ownership (or the desire to own a gun) could be listed as a mental illness.
A lot of people in the old Soviet Union were tried for "crimes against the state" and when not outright convicted, found to be "mentally ill", and sent to "treatment" in re-eduication camps.
Along the the all too real possibility of individuals making the "mental health" determination having personal agendas of their own, there is an ever present problem with any evaluation of mental health.
And that is two fold, firstly, it must rely entirely on the answers given by the interviewee. And second, (and possibly even more importantly) it can all too easily become a "when did you stop beating your wife" situation.
If you don't answer all the questions the "right"way, you are found mentally ill. AND if you DO answer all the questions the "right" way, you are gaming the test, and found to be mentally ill. The real potental for it to all too easily become a lose/lose situation makes any (supposedly) "objective" mental evaluation test for firearms ownership a very, very risky thing.
All else being equal (and it almost never is) bigger bullets tend to work better.