I am not sure why yours is not working but can assure you that my Rachio3 controller does show up in HA. I still use the mobile app since I also have a Tempest weather station and a EveryDrop flow meter and those are integrated well into the mobile app. It works, so I am sure you will figure out what is wrong. I set this up a couple of years ago so do not recall if I needed to do anything special.
I had to use 2 differents integrations.
One for the “Valve Smart Hose Timer” and another one for the “Smart Sprinkler Controller”
Can’t remember which for which (but the Rachio in the HOA integration list, and the ESPHome (I think)). Was not working well.
Then I installed the “Rachio Local Control” via the HACS (The Home Assistant Community Store ( HACS ) is a custom integration that provides a UI to manage custom elements in Home Assistant.)
Also to add to that:
In the Rachio App I see all the devices (Valve Smart Hose Timer” and the “Smart Sprinkler Controller” but on the web page (https://app.rach.io) I see only the “Smart Sprinkler Controller”.
Apparently Rachio focuses on dev for the app. I would not be surprised the web page will disappeared one day.