Glad I am not alone in making the switch.
AGH does have a sort of API that can be used. I actually started working on a driver just so I could turn it off from the dashboard and have it come back on automatically. Otherwise sometimes I turn it off and forget its off for a while until I start seeing ads all over. There is a Pi-hole driver I think on this forum someplace I was using as a starting point.
I don't do much blocking per client, I use the DHCP from the router.
I suspect with Pi-hole that HA was accessing it via command line on the same machine vs using an API? How did it setup the per client blocking? Maybe by adding a removing a group?
Anyway, for AGH, I see in the Persistent Clients section you can use their services blocking per client, is that how you want to block it or would you just do a custom filtering rule? I think both can be done from the API would just be a matter of building the correct command and putting it into a driver.
If you want to check out the API you can find it here and doc suggests to use swagger to view it which works OK. AdGuardHome/openapi at master · AdguardTeam/AdGuardHome · GitHub
I think the last time I worked on it I loaded it in Postman and it was easier to navigate in there.