Anyone with a full Bitcoin node is in theory in significant violation of US law; child porn is a strict liability scenario, where intent doesn't matter.
People have been criminalized for bittorrent, music sharing services, etc. Uninformed people I know hesitate to use Tor clients due to its reputation; well-informed people hesitate to run Tor exit nodes. Why would blockchain get special status?
For example, imagine a blockchain explicitly intended for use by drug dealers to exchange money or CSAM users to exchange materials (maybe they read on HN that it's anonymous, impossible to censor, etc.): Could the government shut that down? Would they? Would you dare allow any such thing on your computer (ignoring that you probably have no interest in either service).
Blockchains allow structure and incentive beyond the data in itself. If someone published all the leaked pii of us citizens from the last decade to the bitcoin blockchain (which would probably exceed high 6 figures in fees,) the financial incentive would effectively permanently protect it. It's about as far from ephemeral as data can get. Governments could hamper the use of bitcoin, shut down exchanges for fiat, but it's effectively impossible to get rid of the data once it's published to the blockchain.
Non blockchain distributed data systems can theoretically be censored, but in the same sense that tor users can be tracked - the resources needed aren't trivial.
Building in a consensus based way of eliminating garbage from non-functional blockchain data might prove legally necessary, as bitcoin nodes are effectively hosting all sorts of nasty and illegal data even now.
Transaction size limits and fees prevent meaningful abuse or use of bitcoin as a data sharing platform, but there are solid cryptocurrencies designed for arbitrary public distributed data. I'd love to see a scihub blockchain, or a better steemit type system.
It'll be interesting to see how cryptocurrencies evolve, and how long it takes governments and cultures to adapt.
What financial incentive? How would that work?