Router Using Backup DNS After New Pi-hole Install

There should not be a "backup" DNS server. Pi-hole should be the only DNS server available to clients.

*** [ DIAGNOSING ]: Networking
[✗] No IPv4 address(es) found on the eth0 interface.

[✗] No IPv6 address(es) found on the eth0 interface.

[i] Default IPv4 gateway: 10.0.0.1
   * Pinging 10.0.0.1...
[✗] Gateway did not respond. (https://discourse.pi-hole.net/t/why-is-a-default-gateway-important-for-pi-hole/3546)

You have a connectivity problem on the Pi. Resolve this and Pi-hole should work fine.