I completely agree. It would be good if the police acquired information on mental health patients – understanding how many individuals diagnosed with psychotic or antisocial disorders in the town were prescribed medication and how consistently they purchased it. Checking the status of institutionalised patients, such as whether they were on leave or if any had escaped temporarily, could be valuable. Premeditated murder seems less likely in this case, as a perpetrator intending to commit such a heinous act would likely avoid leaving evidence. The individual involved, possibly dealing with a severe mental illness, stayed at the scene, displaying disturbed behaviour, and even indulged in some ice-cream (perhaps all he wanted?) It appears his actions were driven by illness rather than calculated intent.
Er, this may not be possible or a good idea for several reasons. Even in a society where mental health is still stigmatized (as it is in Japan), this would clearly be a massive overreach of profiling with uncomfortable ethics. Most mentally ill people (hi, I'm one) are not dangerous; if information about their illnesses are leaked, they could suffer consequences in their social lives for no fault of their own.
The best indication of mental illness leading to violence is, well, violence--which would mean examining a criminal log is far more likely to help lead to a suspect. Unless this person had exactly one exceedingly violent mental health incident ever, which is not how mental illness tends to work, they would have repeated something and been caught.
Especially psychosis. I know the media tells us psychotic people are the most dangerous, and there is some danger, but while someone who is psychotic is capable of planning (it's a misconception that they're not), I'd be absolutely shocked if a psychotic person would keep this secret for twenty years from the entire world. That's just not very realistic. That said, of course you can't rule anything out completely.
Antisocial personality is more likely, but there are some issues with that theory as well, particularly as it pertains to looking at medical records as a possible way to uncover them. If the killer is younger, and the TMPD seem certain he was, then antisocial personalities are likely not going to be diagnosed, even back then. They may well have it, but they wouldn't likely be in a system unless, again, they'd had previous instances of violence against people, which police would likely be involved in. Violence against animals, less likely for police involvement, but also less likely for institutionalization or mental health treatment especially in a society like Japan, where it is heavily stigmatized.
Someone with ASD would be able to keep a secret (very well), but most of them come across as charming and put-together in real life. Hence, they're unlikely to have been institutionalized. Of course there are some that aren't, but being less put together also means that they are less likely to successfully hide things.
The apparent (vast) contradictions between planning and sloppiness do not appear to be actions driven by illness to me, but instead immaturity. Of course, they aren't mutually exclusive, but it feels a bit icky to suggest that mental illness is the primary angle to look at.