The Extensions app was added to GNOME 3.36 3 years ago and is a core app. It has made its way into every distro and it comes preinstalled on all. For example Fedora 32+, OpenSUSE Leap 15.3+, Ubuntu 20.04+, Arch Linux and Debian 11+ all have it in their gnome-shell package. Before that extensions were managed through Software.
I think you mean Extensions app instead of Extensions Manager app? The former is for configuring & updating while the latter is also a software browser and installer. I don’t think a software browser and installer has a place in the settings app. As noted the Extensions app is not optional.
It does if you open the Extensions app; that links to the website and the website informs what is needed to install them. But yeah, they aren’t covered in the Tour app or in the Help and that would might them more discoverable. But would you have found out about extensions if there had been a slide in the Tour app, or an entry in the Help?