As I said before: something is preventing your Pi host device from accessing public DNS servers.
Additionally:
*** [ DIAGNOSING ]: Networking
[✓] IPv4 address(es) bound to the eth0 interface:
10.1.0.10/8 does not match the IP found in /etc/pihole/setupVars.conf (Use IPv6 ULA addresses for Pi-hole)
This can be fixed by running pihole -r choose reconfigure
It is unusual, that router IP, and gateway IP differ. Do you have such a setup?
[i] Default IPv4 gateway: 10.1.0.5
* Pinging 10.1.0.5...
[✓] Gateway responded.
* Received 303 bytes from eth0:10.1.0.10
Offered IP address: 10.1.0.109
Server IP address: 10.1.0.10
Relay-agent IP address: N/A
BOOTP server: (empty)
BOOTP file: (empty)
DHCP options:
Message type: DHCPOFFER (2)
server-identifier: 10.1.0.10
lease-time: 86400 ( 1d )
renewal-time: 43200 ( 12h )
rebinding-time: 75600 ( 21h )
netmask: 255.0.0.0
broadcast: 10.255.255.255
dns-server: 10.1.0.10
domain-name: "WORKGROUP"
router: 10.1.0.1
--- end of options ---