Quite a lot of things can get by without needing any other dynamic libraries other than libc; there are some alternatives to using openssl in Rust that are fairly common. There definitely are places I've seen containers used for building/deploying Rust due to the need for dynamically-linked dependencies, but I've also seen it a bit in places where they already are using containers for everything else, and using the same thing for all of their services regardless of whether they're in Rust, Node, Python can can make the process around deployment simpler rather than having a separate way of doing things for each language.
You don’t need non-libc dependencies to run into issues when copying a binary from one host to another: glibc uses symbol versioning, so (unless you jump through some hoops) a binary built one a machine with a newer glibc will fail to run on a machine with an older glibc.
I ran into this last week — I wanted to install the neovim text editor from the official binary distribution, but because the project’s CI distro has a more recent glibc installed, the executable would fail to run on Ubuntu 18.04.
