But this requires the record to be maintained at a sufficient number of those parties, and that's a problem - maintaining records over long periods of time is hard. This is why creating a cryptographically secured record is worthwhile - it does not require the parties to still be around to vouch for the record.

But that doesn't require a blockchain: each party can individually sign it with their own key, and some third party can collect all the records and the signatures.

Now you have to trust that the third party don't omit anything or add entries from other parties.

There are certainly alternative means to guarantee the same properties, depending on your trust model, but they will quickly start acquiring the same complexities.