If that's true, then why don't people just propose to increase block size?
Because that would be instantly rejected. Many times people hear "you cannot increase block size", but because they have no idea, how to scale things correctly, they cannot see any other solution, so they end up with "obscured block size increase".
We had 1 MB limit. It was reached.
Now we have 4 MB limit, and guess what, it was reached.
After Ordinals, developers will reject any proposal to increase block size, because that limit will also be reached, no matter how high it would be.
While OP is proposing a singular blockchain running along side the mainnet with a (fixed?) block size limit.
a second hashchain that runs in parallel with the blockchain
I cannot find any sentence, where OP would say it would be obligatory to run both chains. In sidechains, you can run mainchain-only, both chains, or even sidechain-only, it depends what type of client you need.
If i understand your idea correctly, 1 block on hashchain linked to 1 block on current Bitcoin blockchain.
There is a difference between obligatory link, and optional link. I think sidechains should be optionally linked, and also through commitments to make it more private. In case of obligatory link, it would be just de-facto block size increase, in the same way as Segwit commitment is.