I've not played with PiHole yet, but other products like it. My understanding is this.
Your Router itself should still use the Public DNS for it's calls. That's how it still sees the internet.
Your DHCP server (which is probably a different section in your router) should be updated to hand out the PiHole IP as DHCP.
What that does is tells any of your devices connecting to the router "Use PiHole to reach the internet", just like you said works when you manually set on your end device.