I think you have the wrong mental model of how the AUR works. You don’t enable it and then it works with pacman (which is what it sounds like you are thinking). Instead, it’s a repository of packages that require a different method of installation. This is described in section 2 of the wiki page you linked:

1. Acquire the build files, including the PKGBUILD and possibly other required files, like systemd units and patches (often not the actual code).
2. Verify that the PKGBUILD and accompanying files are not malicious or untrustworthy.
3. Run makepkg in the directory where the files are saved. This will download the code, compile it, and package it.
4. Run pacman -U package_file to install the package onto your system.

More detailed instructions are on that page.

However, if you want a pacman-like experience, you can install an AUR helper. You’ll still need to install the AUR helper via the steps above, though.

The AUR requires an AUR helper like paru or yay to interact with it that way. Otherwise you can clone the repo using git and make the package yourself.

The AUR is a source-based. You can’t install directly from it. The packages must first be built before pacman can install it.

You might be interested in chaotic-aur. It’s a repo you can add that has a subset of pre-built aur packages that you can install like normal.