No, people routinely write Rust with no third-party dependencies, and yet people do not routinely write C code that is memory-safe. Your threat model needs re-evaluating. Also keep in mind that the most common dependencies (rand, serde, regex, etc) are literally provided by the Rust project itself, and are no more susceptible to supply chain attacks than the compiler.
https://materialize.com/blog/rust-concurrency-bug-unbounded-...
My point is that if you put a very high emphasis on avoiding vulnerabilities, you can either write the code in C with no/limited dependencies (and still risk memory safety bugs), or write the code in Rust with no/limited dependencies and no/limited unsafe code, and get much stronger guarantees for the same/less effort.