Why is it unhelpful to name the political institutions that promote inequality?
If it were as simple as one party promoting inequality and the other party promoting the land of milk and honey it wouldn't be such an issue. But where one side tends to err on the side of withdrawing help from those who need it the other side tends to err on the side of throwing money around like candy and enabling all sorts of unhelpful situations.
Josiah already touched on the whole "give a man a fish / teach a man to fish" issue. Sometimes a need exists that is immediate and urgent such that the only sensible option is to give a man a fish to feed him for today. But unless that man learns to fish he will become endlessly dependent on being given free fish. He might not be in a fit state to learn to fish right now but unless there is some reason why he cannot fish for himself it is unhelpful to simply continue to hand him fish.
Another problem with centrally funded and managed welfare systems is that they are typically hugely inefficient. I'm sure most of us have seen people load up a shopping cart with junk and pay for it with food stamps or similar. Others have seen people loading up with prime cuts of meat, paid for by the taxpayer. For good measure we see people who are unable to afford things being taxed so the money can be handed out so someone else can afford those very same things (whether the things in question are food products, housing, whatever, it's clearly wrong for those who work to be unable to afford the things handed to those who do not, while a part of the reason they can't afford to have them is because of the tax burden).
Locally managed charity maintains a link between the giver and the recipient. If my next door neighbor falls on hard times and I help him out, he is unlikely to repay my kindness by hosting late night parties or vandalising my car. As soon as government gets into the middle and declares that my neighbor has an inalienable right to that help, there is no reason for my neighbor to care about me - I have no choice but to hand over the money that funds his beer-fuelled parties late into the night. Another advantage of locally managed welfare is that it can make provision that actually meets the needs, where centrally managed welfare can do little more than define a need in terms of how many dollars and cents are needed, then hand over the precise amount.
If someone struggles to manage a budget (increasingly common, since home economics doesn't seem to be taught any more) throwing more money at them doesn't necessarily help them - it's better to teach them to budget and understand the ways to make their money go further. Increasing the amount of welfare they receive simply means they are more likely to waste more money and still won't learn anything about budgeting. If someone lives in accommodation that is badly insulated it makes more sense to fund insulation than to give them ever-more money to cover the costs of heating it in the winter. And sometimes people have needs that aren't met by simply handing over dollars and cents - perhaps they need to learn how to prepare a meal rather than thinking they can only cope by buying ready made meals.
For good measure if someone remains on a government-funded scheme for long enough they will resist efforts to change it, effectively becoming institutionalised and unable to see much option beyond continuing to take the free money. Along the way people lose hope, they lose self-respect, they lose purpose. If you compare an inner city ghetto in the west to the slums in the developing world you might initially be shocked at seeing people scavenging to feed their families, and in many ways it is a good thing that we look to avoid that situation in the west. The flipside is that the people who scavenge to feed their families are unlikely to see their communities ravaged by drugs as people desperately try to numb the pain of a pointless existence - you can't afford to sit around getting high when you've got a family that needs to eat.
I don't see any problem with pointing out flaws in particular political approaches to specific problems but merely acting as if one party is exclusively responsible for problems is, in my opinion, unhelpful. After all, if it were merely a Republican issue then one would have thought that eight years of having a Democrat in the White House would have seen these problems all but vanish, no?