It appears that the Android 13 update has removed a new feature found on Pixel handsets that appeared on versions of the Android 13 beta. This was a unified search feature than aligned the search bar found on the bottom of Pixel home screens with the search bar found at the top of the Pixel app drawer. For example, with Android 13 beta 4.1 loaded, both search bars generated results from the web and the Pixel phone.
The unified search bars on the left, disappeared with a reboot after Android 13 was installed