To say this is merely a problem with distribution is absurd.

There are hundreds of thousands of vacant homes in San Bernardino, there are many thousands of homeless that could easily be transported there. If those homeless were to move into those homes they would soon be evicted or arrested and imprisoned. This is an issue with ownership, not distribution. Much food from local restaurants, cafes, and grocery stores is thrown out every day as it nears expiration. Thousands of hungry would love to enjoy this food rather than waste it. This food could be donated or even left out for people to scavenge, yet it ends up in locked dumpsters. Again, this doesn't sound like a distribution problem but instead a problem with ownership.

We do produce more than we need, if need is conditioned on ability to pay.

That's the exact same thing. If we say wealth is not distributed evenly, that's the same thing as saying that only a few people own most things and many people own hardly anything.

Redistributing wealth means spreading out ownership more.

I misunderstood. To say it is a "distribution problem" read to me as if distribution was a verb: that is to say "everyone could have what they want if only the means to move people and things around were available".