Well, after a few weeks of back and forth with Asus, they agreed there is a problem with the router. I'm sending it back to them for further diagnosis. During the sessions with Asus, I did find out that the 50' CAT-5e cable I have going from my computer to the router may be damaged. I'm getting 2-3 times better read speeds using a different CAT 5-e cable (up to 21 MB/second), but the read speeds are still below what I expect. Write speeds are still abysmal at 6 MB/second.
I'm going to be using my back-up router (TP-LINK Archer C5 AC1200) with my new Synology DS118 (I know it doesn't support RAID, but I don't need it) in the meantime. The new NAS, and gigabit switch, should arrive tomorrow. My plan is to put the NAS & switch near the computer in my home office, cable the computer and NAS to the new switch and then to connect the switch to the gigabit router. I'll be using 12" CAT-6 cables to connect both the NAS and computer to the switch. I'll also be going under the house to replace the old CAT-5e cable with a new CAT-6 cable from the switch to the router. When I'm done, I'm hoping to see speeds approaching 100 MB/sec. Since the slow throughput is only really a problem when I access the router's HD from the computer, I think my planned set-up will minimize the speed problems I'm seeing, even if Asus can't get the router working correctly. The other devices I have connected to my home network are mostly streaming, and should function correctly even at the slower (21 MB/second) read speeds.
UPDATE:
I'm still loading data, but my write speeds to the Synology DS118 have reached 87 MB/second (14.5x what I was seeing with the hard drive attached to the router via USB 3.0) while transferring 1.5 TB of files of various sizes. I'll test the read speeds this weekend, but so far I'm a happy camper. UPDATE - Read speeds are 97+ MB/second.
Moral of the story -- Don't expect an Asus router to perform anywhere near as well as a real NAS device. It was definitely worth the money.
Thank you USAFRet and Kanewolf -- Getting a real NAS was great advice!