The assembly language code we inherited from BoringSSL (and OpenSSL) is really important for performance and for safety from side-channel attacks, like timing attacks. I believe that rewriting most of the assembly language code in Rust would be a net loss for security. I have very long-term ideas for how to avoid needing so much hand-coded assembly, but we have higher-priority things to do now. And, the assembly language code is really good. Really.
The C code is increasingly getting replaced (not rewritten or just transliterated) with Rust code, wherever it makes sense to do so. You can see some of the planned work of this type at https://github.com/briansmith/ring/labels/oxidation. To see the past work, review the commit log of ring.
However, we've done tons of work to make the C code safer too. For example, I've written dozens of patches to eliminate cases of undefined behavior and other unsafe coding patterns in the C code. Many of these changes have been integrated into BoringSSL.
Also, we've greatly reduced the usage of the heap. Already, you can use most of ring's functionality without a heap. Importantly, this means that we have solid evidence that, for almost every ring feature, there is zero chance of use-after-frees, double-frees, or memory leaks. It also means that the memory usage is very predictable, which makes it easier to use in constrained environments.
In addition, I've tried to design the ring API very carefully to limit the potential for things built on top of it to misuse the crypto. For example, the API enforces--statically, at compile time--that an ECDHE key can be used only once. Similarly, it enforces--statically, at compile time--that an AES/ChaCha20 encryption key is never used for decryption, and vice versa. Similarly, it ensures that encryption is always properly authenticated--there's no way to accidentally do "MAC before encrypt" and similar things. We even make sure that you don't use less-safe AES-GCM nonces that aren't exactly 96 bits.
Finally, anything that uses ring get all the advantages that come with Rust automatically, such as Rust's use-after-free protection and data race protection. (ring replaced all the C threading/synchronization stuff using the safer Rust constructs already.)
So, even though there is some C code, and even though there's a lot of assembly language code, things that use ring are still getting lots of Rust's advantages.
There are other alternatives that are "pure" or close to "pure" Rust, such as rust-crypto. But, those libraries are missing important things like RSA and ECDH and ECDSA over the NIST P-256 and P-384 curves. That's all needed for a practical TLS implementation.