Definitely not. C was born in a different context, and these days it's not common for any language to have multiple competing implementations, certainly not in its early years. C
needed the standard because there were many implementations, JavaScript
needed a standard because there were many implementations, but Python or C# or Rust or Java don't have a standard (although there are technical specifications of different degrees of rigor) because the standard is what the reference toolset does, modulo what are accepted as bugs by the technical team.
There was no "leadership team" for Rust at Mozilla as far as I know. It was originally a one-person side project like C++ or Python, then it was elevated to an official Mozilla internal project as its potential in the context Gecko was understood by the higher-ups. But again, as far as I know, whatever culture formed around the project did so organically, but also as a conscious attempt to avoid many cultural issues seen in other OS projects. And mark my words, the Rust community as a whole is genuinely friendly and welcoming compared to almost any other internet community of similar extent, and there's nothing sinister underlying that friendliness as far as I can see.
When Mozilla got rid of Rust, the leading technical contributors continued as they had always done (albeit now with considerably fewer full-time paid contributors), as an independent self-organizing entity, but now even less accountable – in regard to technical decisions – for any external stakeholders but the Rust community itself. But some organization was required to foster Rust's growth, to manage all the inconvenient legal things, the interaction with the now several large stakeholders and funders such as Google and Amazon, and so on. So the Rust Foundation was created to manage all that. But the foundation's jurisdiction ends where the technical aspect of Rust begins – all the technical teams are still exactly what they used to be, accountable only to the greater community.
At any point, anyone could have experimented with different implementations with no "committee" saying what to do, but let's face it: first, modern compilers, even simple ones, are extraordinarily complex compared to an early C compiler running on a PDP-11, and second, in light of the first, Rust didn't grow in popularity nearly fast enough for anyone else bothering to write an implementation to experiment on.