Publishing everything on a blockchain means that everybody who’s running a node has access to a copy
I'm not sure that's the case, although the article is rather vague. It says:
[...] the user must register with a node of their choice using their public key. Once registered, users can create channels and invite others to join. Each channel has a separate ledger hosted by the nodes. [...] The data in the ledgers are encrypted, and the secret key is managed by the users of the channel.
IIUC, nodes will not have access to private keys, neither those from users nor those from channels. Users could use their keys to exchange the channel's private key without the node getting to know it. I don't quite understand how user's would exchange their public keys without the node being able to play MITM, though...
Edit: Removed an irrelevant sentence from the quote