"ARCore" menu embeded in "XR" in one project and in "XR Plug-in Management" in another project.

aaand I can’t set up my project to build correctly because of this. Aaaand there is no way to find an answer to this issue anywhere on the web. And yes, I checked the “AR Core” box in Plugin Providers in both projects.

Just cloned the working project, moved asset there and it has this:

which is what is supposed to happen I presume? Yeah clarification: I managed to achieve the problematic menu by upgrading a project from 2018.4.16f to 2019.4.14f.

Nope, that still leads nowhere. Such a beautiful system.

Hey guess what? I upgraded the AR Foundation core and kit and ta-da: another fricking menu!

Your issue is related to AR Foundation/XR Plug-in Management versions. Please pay attention to the currently installed versions.
Also, a different versions of Unity support different AR Foundation versions. If you use version control system, you can find previous versions in Packages/manifest.json file.