Yeah, it has the analogies and differences that you mentioned with CoinJoin, which I didn't know before so thank you. The purpose is not mixing, nor specifically raise the privacy level, even if it could be quite easy to implement such a functionality I guess even if I'm not a fan of absolute secrecy in Bitcoin and think that the "default" system gives users the most reasonable level of privacy.

As for the rest yeah, you got it right. The transaction instances should be delivered to blockchain for validation only when the boxes are full.
The current problem of PoW protocol is that a lot of transactions can't find a block to be inscribed in and has to wait a very variable amount of time to find one.
This L2 network could do this work instead of the blockchain. Boxes are quite like blocks in their functioning, but boxes has different dimensions so that it's easier for the system to allocate transaction instances. Merging transactions in one, so that the the blockchain will see just one big one, is the part inspired by Lightning Network, but yeah, it overcomes the need to find a multisig address.

Also, if you want to make just one single transaction with someone, you don't need to open a channel, with all that implies, nor "hope" that the receiver address is in the channel network of someone that is in your network.
LN still remain the best scaling solution available right now imho, but user-side, I there could be some improvements.