Log rotation to a centralized log server solves that problem, although maybe at the loss of easy real-time access.

Fair enough, I haven't had to deal with complex multi-server application logging.

Your way still works. Syslog can ship over the network since forever.

It's a security pattern to have the logs shipped to another host in production anyway.

I did seriously consider suggesting that (it would be my default approach) but since I don't actually have any experience with big complex multi-server applications, I decided to keep my mouth shut.

There might be a good reason why they don't do it that way and I wouldn't have a clue.