There seems to be a fallacy in your belief that it is the Church which should be the support for the poor.
All Scripture is quoted from the King James translation, unless otherwise noted.
Mark 12:41 through 44 And Jesus sat over against the treasury, and beheld how the people cast money into the treasury: and many that were rich cast in much. 42 And there came a certain poor widow, and she threw in two mites, which make a farthing. 43 And he called unto him his disciples, and saith unto them, Verily I say unto you, That this poor widow hath cast more in, than all they which have cast into the treasury: 44 For all they did cast in of their abundance; but she of her want did cast in all that she had, even all her living.
If the church was to support the poor, how could Jesus be impressed that this poor woman had given her only living to support the church?
Matthew 11:4 through 6 Jesus answered and said unto them, Go and shew John again those things which ye do hear and see: 5 The blind receive their sight, and the lame walk, the lepers are cleansed, and the deaf hear, the dead are raised up, and the poor have the gospel preached to them. 6 And blessed is he, whosoever shall not be offended in me.
Here Jesus is putting preaching to the poor more important than supporting them, and it is not that either God or Jesus himself do not care about the welfare of the poor; but:
Matthew 6:8 Be not ye therefore like unto them: for your Father knoweth what things ye have need of, before ye ask him.
Matthew 6:11 Give us this day our daily bread.
Apparently God wants us to depend on him to provide our daily needs, and that means needs and not wants; however that does not mean that we,who God has blessed so abundantly, should not share with those less fortunate, and may in fact be the reason he blesses us so much, just to see if we are willing to share the abundance he gives us.
Unfortunately we (and I include me) in the U.S.A. have been so blessed by God that we have become greedy and possessive about our wealth. Much of the World doesn't even have a house to live in let alone two automobiles, and only at the most one meal a day let alone three, with snacks and drinks in between. Quite simply we are spoiled by comparison to most of the World.
As Christians should we not take a longer account of ourselves, at least attempt to live as Jesus showed us to do?
Jesus could have had all of the riches and power of the World, but instead numbered himself among the poor and downtrodden.
Matthew 4:9 and 10 And saith unto him, All these things will I give thee, if thou wilt fall down and worship me. 10 Then saith Jesus unto him, Get thee hence, Satan: for it is written, Thou shalt worship the Lord thy God, and him only shalt thou serve.
As Christians we know that what God has offered us for eternity is worth more than everything this World has to offer.
Hope this helps.