If you did that some clever person would set up their PiHole so that their device just always worked, and then you couldn't send them ads and surveil them. They'd tell their friends and then everyone would just use their local devices locally. Totally irresponsible what you're suggesting.
It's a little harder than blocking the DNS unfortunately. But nonetheless it always brings a smile to my face to see that there's a FOSS frontier for everything.
But this new little box would then be required to connect to the home server to receive updates. Guess what? No updates, no worky!! It's a vicious circle!!! Outages all the way down
