One benefit: one could install latest version of Rust on any distribution. Debian stable for instance usually lags behind (with good reason, but still...). I imagine it is mostly useful for checking out Rust though, not for normal use.
What I'm not seeing is the value of a Docker image. Rust doesn't have much in the way of runtime dependencies, which is where the value of a container generally comes in. I don't know of a platform where Docker image would work but rustup.sh would not.
Packages are of course preferable if available and up-to-date.