Did you install Visual C++ Build Tools as well? Without them, things aren't going to work. They easily take up 10+GBs of memory, though - so if you'd rather save up space, consider using msys2 instead.
The page leads you to download vs_buildtools__122341934.1634024943.exe, as I mentioned.
When you start the exe with GUI, it is unclear what is meant exactly with "C++ build tools".
The Microsoft Build tools command line uses "-add" to add components.
As documented here, the component Microsoft.VisualStudio.Workload.VCTools
contains Microsoft.VisualStudio.Component.VC.CoreBuildTools (C++ Build Tools core features)
To answer you question: with the community edition I was only able to install the C++ Build Tools core features.
What am I missing?
Are the core features enough and the rustup-init.exe detection test is wrong?
Not sure about the file that's being downloaded for you, for me it prompts to install the "Visual Studio Installer", which presents the following window. Selecting "desktop development with C++" allows to select and deselect the items that need to be installed - the first 2, along with Cmake, as far as I'm aware, are essential for Rust - and cumulatively they take up almost 7 GB of space.
Yup, worked great, until I got tired of all the space hog for nothing and started relying on MinGW-w64, shipped with msys2, mentioned earlier. In terms of components, it seems like you need these:
Found on the same page you've linked to ealier. And to answer your previous question, the Core features you've tried to install previously aren't enough, from the looks of it.