I suspect you and the parent are using different meanings of the word "structured". They're not totally random or they wouldn't be usable. It's a question of what the structuring principle is.
Am I crazy here? We run all of our app logs and error logs through LogStash and just have a few filters in there to normalize stuff like the timestamp. Honestly the only peace of data that absolutely HAS to be standardized because that’s the piece of data that splits our log indexes, is the primary sorting mechanism, and at what point we roll up an index into some aggregates and then compress and send it to cold storage.