Then, answer the thread question: how important are those programs in alleviating poverty? The answer is, these programs are worthless for that. None of these programs are capable of digging the unfortunate out of their dire circumstances.
None of these government programs can change the class strata, because most people who get money don't know how to hold onto it. People who get rich and stay rich, stay rich because they have the proper discipline in handling and spending money. You can't make the poor rich until they possess that discipline. People need to understand some very basic rules about how to handle money--if they refuse to learn or practice those rules, tough noogies.
What does any of that have to do with keeping old age pensioners, the unemployed and disabled folks able to afford food and a place to live? The bar for the success of income transfers to vulnerable groups is not and has never been "can retirees, veterans, the disabled, and the unemployed become rich through government income transfers because bootstraps". That's just silly, and again, that's your strawman, don't ask others to defend it.
If you want to obsess over the specific term "poverty relief", then actually you've managed to wrongly interpret even the term you've decided to be all pedantic about. These things are pretty clearly successful poverty relief because those groups would be in a much worse situation otherwise.
And mate, if you DO want to play the pedantry game, check out the meaning of the term "alleviate". It does not mean "eliminate", it means "make less severe".