You can’t prove that logs haven’t been tampered with, though. The running system is a black box that could be doing anything to that file before making it available.

This is quite simple. You simply have the running system constantly have the file or data be readable to the public. Now anyone can prove that the logs haven't been tampered with. Even better, you can just use a merkle tree data storage solution like Git to make that verification easy and fast. See Certificate Authority for a fully fleshed out solution in this nature. (I guess you could claim that merkle trees are blockchains, but in that case I would argue that the word "blockchain" is pretty meaningless. Just call them merkle trees.)

But it's turtles all the way down. How does the public know that they're getting the data and not a surrepetitiously modified version? At some point you're relying on the system to be truthful about its contents.

Honestly you brought up really valid points. Thanks. That got me thinking a plenty :)