There's probably some truth to this. But another thing happens. Law enforcement and security personnel spend a lot of time dealing with imbalanced individuals. And that probably colors their behavior and attitude quite a bit.
If relatively low-paid, relatively unprotected hospital workers are "tough" enough to deal with difficult and dangerous hospital patients without using violence, why aren't the police? Unfortunately our laws have made it too easy for police to use violence against people in difficult situations where others in the exact same situation could not or would not use violence. These laws even go so far as to protect police who use absolutely brutal violence to injure or kill individuals who really just need medical care, such as Kelly Thomas [2].
[1]: http://www.thisamericanlife.org/radio-archives/episode/579/m...
[2]: https://en.wikipedia.org/wiki/Death_of_Kelly_Thomas#Incident
All is well (I guess the patient recovered), but I'm not sure she'd call herself "tough enough to deal with dangerous patients without using violence"!
It's not about "being above the law", either, it's about the criminal being better protected by the law, in this instance, than the victim. The policeman likely told her the truth - that no action taken against this guy based on her reported threat is going to deter him from fulfilling it, likely the reverse, so she better be careful.
We used to have sanitariums, now we (non-experts) say it's cheaper/better to just release them into the public. Then get all surprised and pissy when things go wrong.
We've regressed.
Hospital is not a good choice for most people with mental illness. Even people with severe illness, such as psychosis, benefit from good quality community treatment where possible, reserving hospital for last resort intervention.
That good quality bit is important though.
You might want to look at the literature around "Early Intervention in Psychosis" style working.