The timeframes are fuzzy, but it looks like the current Finnish mental health regime was enabled by a law passed in 1990. Since that point, given the 5.6M population times the 214/100,000 rate, we get a total of ~12,000 people committed.
The graph in the linked article shows a reduction in homelessness from about 17,000 to about 4,000, a reduction of approximately 13,000 people.
So I think it's fair to say that Finland's mental health changes have been responsible for the overwhelming majority in the reduction of Finland's homelessness problem. This is consistent with the point that I was trying to make elsewhere in this thread [1].