Why does the blockchain need to be a single chain?

One of the big issues plaguing the blockchain is throughput - since all the transactions go into a single ledger AFAIK, the transactions need to be serialized.

But why does this need to be the case?

Imagine you had a blockchain you could fork at arbitrary points, let's limit ourselves to 2 levels - a core level and a sub level.

On the core level the only thing you can do is create sub-chains, let's say through a PoW algorithm.

On the sub-level, it would work like a regular blockchain, so you could write transactions into each chain with a PoW algorithm.

Let's identify each user with, a letter, and each subchain with a number.

Now, the problem would be that wallets could be distributed between chains, so for A, you'd need to add up the balance of A0 and A3 if we presume that A has coins on these chains.

Now, if A wants to transfer money to B, let's say she transfers from wallet A0 to B1, she just writes a transaction into subchain 0's ledger to transfer coins from A0 to B1, with the usual PoW algorithm. Then let's say the miners who manage chain 1 record this transaction into it out of charity - since A has PoW already invested, these sort of cross-chain transactions can't be spammed.

Now I described the algorithm with 2 layers, but it could scale to an arbitrary number, to minimize the chain lengths, as long as the chain id's are unique.

Now I'm pretty sure I missed something dumb, but I can't see it :)