Here's some more interesting info: The last syslog entry for my RT-AC88U was on the evening of July 6. The router was configured to automatically reboot each evening at 3AM. There were no syslog entries that this occurred on the morning of July 7.
This afternoon (July 8) the router was still functioning properly, except that I couldn't successfully login to the GUI. I did manage to successfully connect and disconnect OpenVPN connections to the OpenVPN server on this router from another Asus Merlin router, and was able to send traffic over the OpenVPN connection. Again, there was no record of these OpenVPN connections in the syslog of the router.
Finally, this evening, I logged onto the router using SSH and issued a REBOOT command. At that point the router hung for hours, until I finally power cycled the router, at which point it successfully rebooted. I subsequently upgraded the firmware to 384.18. I'm keeping my fingers crossed that this will help, but I am not particularly optimistic.
If anyone is interested in looking at my syslog file, I have saved a copy. Most of this is greek to me.
This problem really needs to get escalated and fixed. I am using these routers to connect multiple remote offices and I can't handle these types of reliability issues. What is really strange is that some of the routers in my network are having this sporadic issue, while others are solid as a rock.