Note that you said asymmetric cryptography and your parent said that asymmetric cryptography is not the basis of the block chain. And they would be right, at its core, the blockchain consists of blocks of transactions in a Merkle tree. The Merkle tree typically uses a cryptographically strong hash, otherwise it would be possible to modify the Merkle tree without anyone noticing. But a hash is not asymmetric cryptography.

In many applications of the blockchain, asymmetric cryptography is used to sign transactions. But that is largely orthogonal to the blockchain as a data structure. The blockchain as a data structure has useful applications without transaction signing, etc.

What makes the discussion clouded is that some people define 'block chain' as transaction blocks in a Merkle tree, while others also consider distributed consensus, etc. to be part of the block chain.

Blockchains as defined this way have existed long before there was a proof-of-work based consensus protocol to make Bitcoin a reality. In their more general form they are called "Merkel trees". Filesystems aren't generally structured this way but, for example, git repositories are Merkel trees.

The only cryptography is the SHA1 hash that links commits together, otherwise all cryptography is optional.

And that's totally okay because such repositories are usually run by people who have the authority over it.

A blockchain is a special kind of linked graph that uses cryptography to link each node, otherwise, there are no additional requirements and there are plenty of reasons you might not want anything else.