There's a certain pride in being able to provide for yourself and your family without depending on the charity of others. That's very ingrained in middle-class US society, and back in FDR's day that pride was strong among farmers, poor immigrants, and most others barely managing to scrape by through the Great Depression. Those are the folks I believe FDR was talking about in your quote.
But below them are poorer people, desperate people, people who are still struggling to provide for themselves but are loosing ground because the odds are against them. (eg: everything is more expensive when you're poor, opportunities pass you by, etc.) In my opinion, the goal of income redistribution should be to lift these people up enough so that their struggle to hang on is changed into a slow climb towards independence. They don't need riches for that, they just need a fair playing field and enough wealth to stay in the game. No one in this group, I believe, would be hurt by the assistance.
Now, there is a group even further down the ladder, who have given up the struggle, and who couldn't care for themselves even with adequate wealth. These are the people that opponents of various income redistribution proposals use as the examples of "freeloaders" who will "take all our wealth" to get a "free ride". It's disingenuous to highlight this group while ignoring the group I described above, but I do have to agree that this group would not be helped by income redistribution. This group needs a better version of the social safety net that we already have in place: shelters with free meals, medical care, drug addiction treatment rather than incarceration, education. These people need help learning to become productive members of society before they can benefit from directly sharing in the wealth of society and move on to helping to produce that wealth.