The supply chain attack always existed. C because it didn't have a package manager made it slightly harder in that a dependency wouldn't be automatically updated, while Rust can do that. However this is very slight - in linux many people use libraries from a package manager which gets updated when there is a new release - it wouldn't be hard to get a bad update into a package (xz did that).
If you have packages that don't come from a package manager - windows install, phone installs, snap, docker, flatpack, and likely more you have a different risk - a library may not have been updated and so you are vulnerable to a known flaw.
There is no good/easy answer to supply chain risk. It is slightly different on Rust because you can take the latest if you want (but there is plenty of ability to stay with an older release if you want), but this it doesn't move the needle on overall risk.