If BF4 was written in Csharp, or java (or rust or go?? I'm sure it would still have just as many bugs. One of peoples biggest complaint is the kill shots that you don't see, but that's a design choice (client side hit detection).
Of course, it's impossible to speculate without seeing the codebase, but considering that Rust makes several classes of C++ bugs impossible at compile time, I'd be hard-pressed to imagine that a Rust version wouldn't be less buggy.
If the safer type system gives the devs an unwarranted sense of security, they might write less tests, be less careful in their design, or wait longer between audits and other sanity checks.
If on the other hand the devs understand which classes of bugs aren't ruled out in Rust, then sure, you will end up with fewer bugs.
Rust's type system eliminates the need to test for whole classes of bugs, because they are statically checked for at compile time. This means that tests can be more focused on logic errors rather than standard book keeping. If you look at the example set by the rust repository itself (https://github.com/mozilla/rust/), it is heavily tested and every single PR (https://github.com/mozilla/rust/pulls) is reviewed before merging. This discipline definitely filters down into third party libraries.
