Incrementing an atomic counter every time a reference is copied is a significant amount of overhead, which is why most runtimes prefer garbage collection to reference counting (that, and the inability of referencing counting to handle cycles elegantly).

Rust doesn't rely on reference counting unless explicitly used by the program, and even then you can choose between atomically-reference-counted pointers (Arc) vs non-atomic-reference-counted pointers (Rc) that the type system prevents from being shared between threads.