OK thanks. There is no reason your AppleTV should not use the Time Capsule's Ethernet port. According to Apple, if the AppleTV detects a viable Ethernet connection it will use it in lieu of an equally viable Wi-Fi connection.
Use the same Ethernet cable to rule out a potentially defective one.
Try "forgetting" the AppleTV's network settings:
Apple TV (2nd and 3rd generation): Troubleshooting Wi-Fi networks and connections - Apple Support
If that doesn't work, reset the AppleTV's settings to their factory defaults:
Restore your Apple TV (2nd and 3rd generation) - Apple Support
You would need to configure it again.
If that still doesn't work, it calls the TC's Ethernet ports into question. I understand you already addressed that by substituting other devices, but that's the only remaining potential culprit. Troubleshooting the TC essentially involves a "hard reset", which does not affect its backups.