This whole thing really isn't that complicated, I don't know why it's such a contentious point....
1) When the economy contracts, the poverty rate increases because there is less money being paid in wages.
2) When the poverty rate increases, it can cause the economy to contract because fewer people are spending money beyond the essentials (assuming they can even spend on that).
3) Due to 1 and 2, a contracting economy and increasing poverty can lead into a self-perpetuating downward spiral.
4) Due to 3, a system is needed to break this cycle and ensure those in poverty have at least a minimum amount of spending money. Government programs provide this safeguard.
5) Due to 4, taxation is necessary to offset government spending.
6) Due to 5 and the fact that an expanding poverty rate is part of the hypothetical problem, it doesn't make sense to tax the poor or the middle class who might be pushed into poverty by that taxation. There is only one group left able to bear the burden in this case: the wealthy.
7) When the economy contracts, even those who were well-off before can be negatively affected.
8) Due to 6 and 7, taxation of the wealthy to provide basic benefits to those in poverty can aid in allowing the wealthy to remain wealthy, when they might otherwise fall into the middle class or lower during an economic downturn.
What's so hard to understand? Maybe in good times the equation is different, but government social programs are not about good times: they are about mitigating how bad the bad times can get.