A very strange thing happened to me.
ArcGIS Server appears to be working fine, but the log is not written to the filesystem.
I also checked the folder and it is out of date.
the only thing I can point out is that there are many published services.
Yesterday ArcGIS Server crashed. I realized that there were about 270 ArcSOCs, so I converted all the services from Dedicated to Shared Istance, now there are about 50 ArcSOCs on the server.
As "normal" services there are about 250 in shared instance, while hosted services are about 1430.
Could you double check the directory the logs are being printed out too? If you are able to take on some downtime, running the Configure ArcGIS Server Account tool is my go to to try to resolve this. I would also check to make sure that this directory is not read only.
Currently I find it difficult to stop ArcGIS Server to run the tool.
I can tell you that for the sake of scruple I performed the change of the log directory and the creation of the folder works correctly.
below is a screen where you can see that the user zgis ... has full control of the folder.