This is a human organization problem that could be mostly solved by technical means. For example, a way to notify big nonprofits that give things to the homeless and ask them to come to a specific location at a certain time to get whatever they want.
I think the re-use people were likely honest when they said nobody wanted that stuff. I don't think there's a magic technical solution.
I suspect too that the needy and poor are really the last people with the room and time to stash away a bunch of things they're not using but maybe one day someone will show up and install them... and hopefully they still work.
I don't know. It's probably a problem of effort -- if you need a sink, do you buy a new one, or spend tons of time perusing various used-stuff markets looking for something suitable? There may be an opening there for technology to solve it.
Sinks generally don't just break. If you're buying a new sink, it's likely to replace a perfectly good used one. So having an ability to easily find even more used sinks wouldn't really change much. Never mind if you're going to spend the effort/money on the work of swapping a sink, you might as well pay a little more for a new sink.
ReStore has their niches, but overall there's a large feel good hopecycling dynamic as well. So many used furniture sets just sitting around indefinitely.
Exceptions of used things people seek out basically prove the rule - things that you can't get any more (solidly built, repairable appliances, wide lumber, etc), or vintage styles coming back around into fashion.
I think we just identified the problem: people who are homeless would really benefit from being able to make food, but they can't plug in a stove no matter the cost.