How, exactly, are side chains like Polygon related to Ethereum, other than using an ERC-20 token?

I understand the idea of roll ups to scale Ethereum by rolling up many transactions into one and using the security of the Ethereum Blockchain, but I don't understand how side chains are really Ethereum L2 solutions. If all transactions are taking place on a separate Blockchain and never being recorded on the Ethereum Blockchain, how is that scaling Ethereum? If Ethereum were to disappear, the value of Polygon might go down, but it seems to me it would still be able to operate as its own Blockchain. It's just profiting off of the Ethereum name while having almost nothing really to do with the Ethereum Blockchain.

That said, I've heard that the Polygon network has started integrating roll-up L2 solutions too, but let's ignore Polygon specifically and just focus on the issue of sidechains as a scaling solution. Can anyone explain to me how they are actually scaling solutions and not just parasites?