The Journal - a proposed syslog replacement

Nope. But note that a git repo that isn't cloned anywhere would be vulnerable to an attacker simply rewriting the hashes and thus being able to alter a repo. On a modern CPU hashing is fairly fast so any project that isn't as huge as the kernel could be compromised. That isn't a problem because anything important is replicated. That is the key to security, the hashing just improves it.

For syslog, replication alone is enough to solve the problem. Adding crypto foolishness and a bunch of binary fluff just makes it more complicated and reduces security. Put a log server somewhere on your network with only the syslog port open. If you are really paranoid you could store sha256 sums of each log as you rotate and pack it away on yet another machine or better on paper. Or just log the important entries on a line printer in real time as others have already suggested. Use a line printer without reverse paper feed and it is physically impossible to change the permanent record.