There was nothing in the article to indicate any of your assumptions are correct.

Mental health is not the only reason people harm other people.

There are plenty of instances of people killing their ex, their ex's significant other, and then themselves. Trying to take everyone else out before taking yourself out is an asshole thing to do whether it's a mother, ex, or recently fired employee.

Dark triad tendencies aren't always the result of a lack of mental health options.

There was a dad who superglued his daughters eyes shut. Clearly there was something wrong in his head, but he was also an asshole at the core. Resources are better spent on the dad who let's his kids eat all the ice cream they want.

Husband/boyfriend/mother/father/sister/brother/church/friend/stranger could have helped her if she allowed it. If she insisted on being in control noone could ever help.

It keeps happening not because of lack of resources, but because lack of willingness to accept that poor decisions have consequences.

Maybe we should create some agencies that study the effects of single parenthood so that we can gain a little more insight.