Yay is a 3rd party package manager. The 1st party package manager does not interact with the AUR.
Yay itself is in the AUR. You have to go out of your way to install it.
The Archlinux docs on AUR helpers lead with a red warning: https://wiki.archlinux.org/title/AUR_helpers
Oh, I thought it was a package from the repo. (I didn’t use any of those third party package managers, just stuck to manually doing everything when using the AUR, which was fine because I used it sparingly).