it would be interesting to see if there was increase in involuntary hospitalisation and that’s what actually made the difference

No, I don't think that the establishment of crisis lines or putting more resources to work, will result in such an increase. It is a question of moving the funnel and allocating better resources.

The MH Crisis Line may prevent unfair arrests and jailings. It may prevent certain altercations with law enforcement. It may prevent, or at least accurately predict, domestic violence incidents and so forth. The problem with 9-1-1 is that calling for an Emergency resulted in the dispatch of armed police and/or paramedics and firefighters who were poorly equipped to deal with the autistic ADHD non-verbal manchild having a meltdown. Also, many communities are filled with hatred for cops and other first responders in uniform. Sending them can cause secondary incidents and violence.

So if you've got a Crisis Line with people equipped for mental health stuff, then you can send the correct responders. Many municipalities have already established teams like in a "Care Van" who can connect with citizens, establish rapport, and get them referred to services, non-violently, but really urgently.

That will make all the difference. Perhaps it will result in more, or fewer, involuntary hospitalizations, but it represents a solid funnel into those services and allocates more resources to deal with incidents that would only be exacerbated by armed and militarized police/fire/EMTs.