With a hash chain[0], an idea from 1981.

[0] https://en.wikipedia.org/wiki/Hash_chain

I am interested in this and I have an emotional desire for you to be right; I'm skeptical of every application I've ever seen for blockchain.

That being said, it is not clear to me how a hash chain guarantees that the data it stores hasn't been modified.

Okay, maybe I should have posted the link to the Merkle tree page[0] instead.

The basic idea is, you have a chain of 'transactions' (like in bitcoin or think git commits) und you store the hash of the content + the hash of the member before as a checksum. This way, if you alter the log file, the hash for the modified entry is suddenly different, and the chain breaks, because the next member used your hash as input to calculate their checksum. So you can not only see that the file was tampered with, you can also say which entry was modified.

Edit: Merkle trees[0] generalize this concept into trees, which are e.g. used in ZFS (the file system), git and in the blockchain.

[0] https://en.wikipedia.org/wiki/Merkle_tree

It seems to me that if you have a copy of the information locally and compare it to the remote, you can see whether they diverged with or without a hash - the hash certainly makes it faster, but it's not clear to me that a hash chain or a Merkle tree offers a guarantee that's as powerful as the blockchain's promise.

(Social factors and forks to change our minds on blockchains notwithstanding)

Like git. The value isn't necessarily in that there can't be conflicts, but that conflicts are straightforward to surface and resolve.