Thanks a lot for the prompt answer !
I actually thought of a similar issue but, it seems my router does not allow for different DNS configuration between WAN and LAN sides. It appears very similar to this post I just stumbled upon while researching my router model
I bought the router close to 10 years ago now. Maybe it's time I change it for something that would allow me to use PiHole as I intend. I made PiHole work on it but it was years ago and it is likely that a firmware update created this blocking point.
Maybe I can just try PiHole to act as a DHCP/DNS and only use the router for WiFi before buying a new one.