Yeah, I believe that's the reasoning too. It was probably sensible when the vast majority of people were members of some organised religion or other, and the churches did play a significant role in looking after the poor and needy. But nowadays, neither of those things is true.
You show a lack of knowledge. Obviously you have not heard of, Anglican Care Network, Presbyterian Support New Zealand, Catholic Social Services, Foodbank, Lifewise Charity of the Methodist Church, Baptist Community Ministries etc. Take a look here and you might learn something. http://nzccss.org.nz/about/members/ These are just the Christian organisations no doubt there are non Christian ones as well.
There is also plenty of organisations like Vinnies providing emergency accommodation, locally they also provide 1000 sandwiches a week to Hamilton school children
I suspect the Sallies would get a fair amount of their money back, whereas the Catholics would disappear their profits overseas.
A wildly inaccurate and insulting assumption.
Most main stream churches struggle financially. I very much doubt the Catholic church or any other church (Anglican, Presbyterian, Methodist etc) has any spare money (profits) to send overseas. Yes, many of then have same nice looking churches, but you'll find these mostly were built with donations and bequests. They struggle to find the money for their upkeep and rely on generous benefactors.