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Assuming all assets get stored on S3, the other common writing scenario is logging. How would multiple machines writing to the same log file work?

Generally, you wouldn't write to the same log file. Prepending/Appending the instance name to the file would be enough to make it unique.

It wouldn't. NFS is not fully POSIX compliant, largely due to not being cache coherent. In particular, O_APPEND is not supported.