Rustls still outsources cryptographic primitives. I believe the currently supported providers of those are… drumroll… AWS-LC and Ring. The latter is a fork of BoringSSL. The article describes AWS-LC and BoringSSL as "Googled and Amazoned to death; they don't care about anyone but their own use cases".
The primitives aren't a problem. You can't write them in any vaguely modern high level language. And when I say "High level" I mean that the way K&R does when they describe their new C programming language as high level. The reason you can't write cryptographic primitives in a high level language is that optimising compilers love clever tricks which offer data dependent performance, across every layer of their design - but in cryptography we want constant execution time regardless of either the plaintext or keys used.
The problem with OpenSSL isn't these cryptographic primitives, that's why you will see basically the same primitives re-used in lots of different places. It's like finding out that the guy who was just arrested for murder also eats pizza. Yeah, people do that. The problem wasn't the pizza, it was the murder. OpenSSL's implementation of the AES cipher isn't broken, the problem is elsewhere.
Also, even if rustls is using aws-lc-rs, you still get the TLS parts from the rustls project, and aws-lc-rs is just lower-level crypto. That means there's less places for Amazon to say no; they either implement an algorithm or don't.
There's even a project for a deliberately OpenSSL drop-in compatible Rustls backed library. It is intended for specific projects because OpenSSL is sprawling and they don't implement most if it, but in principle if you use the same parts of OpenSSL your C likely works with this safer + faster alternative today, why not recommend it to your users.
There is a rustls side project called Graviola that's building a fast crypto provider in Rust+ASM. It's taken an interesting approach: starting with an assembly library that's been formally proven correct, and then programmatically translating that into Rust with inline assembly that's easy to build with Rust tooling.
The state of things sucks :-(