Why doesn't each node store only a portion of the blockchain?
This has been proposed before and is possible with Bitcoin, but it's not clear how it would be executed. The blocks aren't an important part of a running node, for most people you can throw them out as soon as you have processed them into your UTXO database.
The problem comes when deciding which parts to store, and signalling to other people in the network that you have a certain subset of blocks. You can't advertise a list of hashes as that would be massive and inefficient (many megabytes for each peer), and making a deterministic random selection is extremely problematic. Choosing ranges of blocks is not ideal because the sizes are not consistent, 1,000 blocks at the beginning of the chain is servers orders of magnitude smaller than ones at the tip. In a naive implementation of random selections peers attempting to sync with the network may need to make thousands of connections to find a single peer that has a specific block, which is obviously not feasible.
From my understanding of the blockchain, let's assume a particular node starts at the genesis block. Let's assume another node stores 1% of the blockchain past the genesis block. Couldn't multiple node verify block hashes for their shared 9% at a rate of once every 10 minutes? It wouldn't really use that much data -- you could hash the 9% of the blockchain to a particular value, produce a public key, and see if the other nodes can match a private key to said public key.
This is not necessary and highly vulnerable to Sybil attacks, you don't need to verify blocks once you have seen them. Pruned nodes work like this today, only they throw away every block and store absolutely nothing more than they are specified to sore. They don't advertise which blocks they have because there is no mechanism for that.
You might want to read up on how the autoprune patch works and get a better grasp on the threat models that need to be tackled here. You've missed some of the operation slightly, note that pruning for nodes nodes does not operate in the way described in the white paper, the UTXO is a newish concept added in 0.8.0 era Bitcoin Core.
A related question: Is this what Electrum does?
Not at all, Electrum servers are a full node and an extremely heavy address based index, almost doubling the storage size. The client is light weight but the sever most definitely is not. There's no sensible way of maintaining these indexes in a sharded way, though you could break addresses into multiple pools and hope that people maintained enough of each shard for clients to be able to request subscriptions for all of their addresses. Ideally systems would be designed that don't require full indexes as they are getting increasingly unwieldy to work with.
