> Does IPFS actually solve the problem they set out here though?
No, none of these distributed P2P networks (that I've seen) do. The problem isn't just building a DHT (kademlia-based networks have existed for years) the problem is incentivizing people to seed - ideally people with high-bandwidth and massive amounts of storage who are seeding data that people want. In other words you need to build an economy on top of the network.
Cryptocurrency could be used for this, so long as its a cryptocurrency that supports instant micro-transactions (ie. you don't want to be writing to a blockchain every time you download a 10KB gif). So maybe someone will get around to building an IPFS-clone but on top of bitcoin's lightning network or something like that. Clients would need a way to decide which peers to send requests to based on who's offering the best speed/reliability vs price. Servers would serve higher-paying requests with higher-priority and would drop any requests that are too stingy to even cover bandwidth costs. Using the network wouldn't be free but it would be extremely cheap and fast and reliable.
One problem this doesn't solve is getting other people to backup/seed your data for you. An idea I think would work for that would be a prediction-market based reputation system for peers acting as storage hosts. That is, peers could advertise themselves as storage hosts, you could upload your data to them (for a fee) and they'd give you a cryptographic receipt promising that they'll still be able to deliver the data at some later date. People could then make bets on whether a host will fail to uphold any promises before a certain date, and the betting odds would be a measure of a hosts reliability. Clients that are uploading their data to the network would take that reliability measure and the price into account when choosing hosts to upload to. At any point a client could publicly challenge a host to provide proof that they still have the data, and if the host fails to provide proof the bets would close in favor of the punters who betted against them. Otherwise, once the bets expire, they close the other way. This would all need to be built on top of a blockchain though you couldn't use bitcoin for this until/unless they add support for covenants or sidechains.