A proposal for a genuinely distributed blockchain and pruning
One thing you're missing is that nodes don't actually need access to blocks.

The only time a "scan of the blockchain" occurs is to give historical blocks to another node that is syncing, or to find historical transactions a wallet missed. Solutions for those case are not hard. Sequential bulk storage and bandwidth are cheap and it's not something every node on the internet needs to provide to everyone. You can find movies on the internet for free that take not much less space than the block chain. 

Modern full nodes can even run without keeping blocks on disk at all (see the -prune option in Bitcoin Core). In this mode, all blocks are still downloaded and verified one by one, but not kept on disk after verification. The only thing that is kept is the UTXO set - the database of the current state of the ledger, rather than its entire history. This is sufficient for validating future transactions and blocks, participating fully in the P2P network, and provide wallet functionality.

So can't we use your idea for the UTXO set instead then? Maybe, but it's pretty pointless. The reason you run a full node in the first place is to be able to verify that nobody is violating any of the rules. If you're willing to not do that completely anymore and instead start relying on trusting some of your peers, why bother? Bitcoin provides a lightweight mode where you don't need access to either the blockchain or the UTXO set at all.