I am nodding my head. There is a difference between being fundamentally troubled and hopeless vs being cold-hearted and vindictive. But I also wonder if a person could be both. (When evil people lose their minds.)
IMO a person can be troubled and feel hopeless, but still love and care for their child/ren. Many parents struggle with divorce, financial hardship, grief, In-Laws, family tensions, but they don't murder their child/ren. On the other hand, some people may have an underlying cold-heartedness which has been masked, then following stressful events, they may act out their vindictiveness on their child/ren.
The case here may involve both mental health and legal assessments to determine the degree of 'mad' and 'bad'.
1) Was the murder caused by underlying 'madness' i.e. major mental illness which rendered the murderer 'mentally incompetent' at that time?
2) Was the murder caused by 'badness' i.e. a preexisting character/personality disorder, with a desire to punish the other parent and a desire to get rid of the child?
These things are yet to be determined.
Thankfully CCTV captured the actions of the murderer in this case - without which this case would be much more difficult to investigate.
The attempt to cover-up the murder and in addition, contrive a scenario to 'blame' someone else - looks ominous in this case.
My opinion only.