TIL Rust is working on its own version of ASAN and UBSAN: https://github.com/rust-lang/miri How shocking.
I guess the problem is "unsafe" blocks. Someone else can probably elaborate better, but if you have pure Rust code with no unsafe blocks, you shouldn't have any memory errors. But large systems written in Rust might still benefit from finding memory errors dynamically rather than statically with something like ASAN.