Most C binaries usually do not contain everything needed for their execution. It would make them too platform-specific. What most c programs do is that they use standard c library from platform for low-level things and communication with the system like memory allocation or stdin/stdout things, for example.
It is easy to install another shell indeed, but it is quite difficult to configure it. While installation of DE is usually done with just one command. And you can use linux without DE, but not without shell. Many distributions even do not install DE by default at all.
If you want serious optimizations - then Gentoo is your choice. But seriously, there won’t be any serious difference between distributions. What really matters here are DEs and browsers. I would recommend some kind of lightweight window manager like i3 or dwm. If you do not want to configure everything yourself, then your choice is lxde/lxqt. Also, you can use distros without systemd (void, artix, devuan, gentoo etc), but that does not matter that much.