Church Giving to the Poor

Here are case examples of churches giving to the poor, which may prove a useful reassurance to your question of "why does this not happen nowadays?"

I was a member of a Lutheran congregation that had a budget commitment for "Benevolence" (which was everything from running the national denomination's HQ to missionaries to Lutheran Social Services). The congregation also gave another 10% of its budget to "Social Ministry", which was distributed to a number of local and national direct-relief charities for the poor. That congregation also had budgeted a few hundred a year for a pastor's discretionary fund, mostly people for people in trouble who'd stop by the office ("I ran out of gas", etc.).
At the same time, both that Lutheran congregation and my current one (a non-denominational evangelical one) have regularly held special offerings for a member of the congregation in trouble. Usually a medical emergency that devastated the family's finances.

As far as whether Christians should "give up all they have", that passage is about challenging one's faith: is it in your own works? There are two passages that should be thought about in relation to each other.

Luke 19:8

But Zacchaeus stood up and said to the Lord, "Look, Lord! Here and now I give half of my possessions to the poor, and if I have cheated anybody out of anything, I will pay back four times the amount."

Matthew 19:21

Jesus said to him, “If you want to be perfect, go, sell what you have and give to the poor, and you will have treasure in heaven; and come, follow Me.”

In the first case, Zaccheus tells Jesus that as an act of contrition, he's voluntarily giving half to the poor, and is commended by Jesus. In the second, the rich young ruler first lies to Jesus by claiming he has kept God's law since his youth. Confident that his level of obedience has justified himself (has earned salvation), is told that the requirement is to give all to the poor. Jesus forces the ruler into a situation where he has to either go away or seek out some other source of righteousness (i.e., a source which comes from faith in Christ). Other passages tell us to give to the poor (Deut 15:10-11) and to him that asks of you (Matt 5:42). These are literal commands of Jesus: obey them, whether the person has short term need or is destitute. Today, we would remind anyone confident that their level of economic fairness or social justice has earned favor with God that the requirement is to give all (Matt 5:42). To the one in despair over their sins, they are reminded of God's free, unearned grace to those who believe.

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.

and

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.

This is largely going to vary on a per-local-church basis, but in addition to Affable Geek's answer, some churches have chosen as a body to focus their efforts in a specific area (perhaps a verbal/written commitment to a particular missionary or translation work or homeless shelter etc) - and it would be a violation of their word to renege on those obligations without good cause (eg, having lost the funding themselves).

What I write here is too long for a comment, though admittedly not a complete answer.

Many churches do help people financially. Mine has certainly done so, and I'm pretty sure it would do so for me if I needed it. Fortunately I have substantial savings, and insurance, and live in a country with a good welfare system, so I would expect that I would never need it. The same is true for almost all of the members of my church, and most churches I am familiar with. The few people it doesn't apply to ARE taken care of. Churches that I know of in places where there is not a good welfare system and much more widespread poverty frequently care of their impoverished members.

Denominations such as Old Order Mennonites make an explicit policy to practice this in a complete form. Members of those communities do not take insurance, and opt out of welfare and healthcare insurance programs, because the community will take care of any members who suffer disasters. Hutterites go a step further and practice communal living and sharing of property.