Edit - this appears to be a possible firmware issue with my router (Netgear R7800). It's obviously not an issue with the pi-hole. Thank you for your help!
Is this limited to a single client, or are you seeing this on more than one client?
Your debug log shows that Pi-hole is working properly.
From both the client (not via ssh session to the Pi) and from the Pi terminal (run each command on each platform), please post the outputs of the following:
PS C:\> nslookup pi.hole
Server: UnKnown
Address: 192.168.1.1
DNS request timed out.
timeout was 2 seconds.
Name: pi.hole
Address: 192.168.1.69
PS C:\> nslookup google.com
DNS request timed out.
timeout was 2 seconds.
Server: UnKnown
Address: 192.168.1.1
DNS request timed out.
timeout was 2 seconds.
DNS request timed out.
timeout was 2 seconds.
DNS request timed out.
timeout was 2 seconds.
DNS request timed out.
timeout was 2 seconds.
*** Request to UnKnown timed-out
PS C:\> nslookup google.com 192.168.1.69
DNS request timed out.
timeout was 2 seconds.
Server: UnKnown
Address: 192.168.1.69
DNS request timed out.
timeout was 2 seconds.
DNS request timed out.
timeout was 2 seconds.
Non-authoritative answer:
DNS request timed out.
timeout was 2 seconds.
Name: google.com
Address: 216.58.194.206
Edit - I'm looking more into the router angle. Since you said my debug looks like it's working properly (I had no idea, which is why I posted here), and manually setting my PC's DNS to the pi hole seems to function correctly, I'm going to focus on the router. Thank you for the help!
If they don't play nice, then use Pi-hole for DHCP and disable DHCP on the router.
If you can't disable DHCP on the router, then try restricting the IP range of the DHCP to a single IP address (the IP of the Pi-hole). Then enable DHCP on the Pi-hole on a non-overlapping IP range.
If you want to keep using the router DHCP, then start with disabling IPv6 on the router. IPv6 can result in DNS bypassing Pi-hole.