When a mentally-unstable person uses the courts improperly, the court can label them "vexatious litigant" [0]. Then they must receive permission from a judge to file any new lawsuits. Courts don't do such things quickly.
It's not the job of courts to force people into the care of the a mental institution. The government must petition the court for permission to do that and show that the person is a danger to themselves or others [1]. Sometimes a court will recommend mental hospital rather than prison for a convicted criminal. This Craig Wright person doesn't seem to be using violence on himself or others and hasn't been accused of a crime. Therefore the system will not force him into a mental institution.
It's not the job of courts to force people into the care of the a mental institution. The government must petition the court for permission to do that and show that the person is a danger to themselves or others [1]. Sometimes a court will recommend mental hospital rather than prison for a convicted criminal. This Craig Wright person doesn't seem to be using violence on himself or others and hasn't been accused of a crime. Therefore the system will not force him into a mental institution.
[0] https://en.wikipedia.org/wiki/Vexatious_litigation
[1] https://en.wikipedia.org/wiki/Lanterman–Petris–Short_Act