Given that C was the dominant programming language for UNIX applications for over a decade, I think we can look to history for an answer to this question. And I believe the record shows that the answer is "no."
The way these things usually work in practice, the evidence that a new paradigm improves things usually builds up slowly. There will never be a point at which someone proves mathematically that C is obsolete. Instead, the gentle advantages of other options will get stronger and stronger, and the effectiveness of C programmers will slowly erode compared to their competition. At first only the people who are really interested in technology will switch, but eventually only the curmudgeons will be left, clinging to an ineffective technology and using bad justifications to convince themselves they aren't handicapped. Is Rust the thing that everyone except the curmudgeons will eventually switch to? Who knows, but if you don't want to end up behind the industry then it might pay to try it out in production to see for yourself. If you don't make room for research and its attendant risks you will inevitably fall behind.
Is this a rhetorical question? Clearly the answer is no.
It seems like there is a big difference between a "mathematical certainty that certain bugs will not occur" if you play by the rules and just be a better programmer so that you don't write errors. Not that you won't have bugs in Rust but it seems like we should move towards having our tooling do more of the heavy lifting in ensuring correctness. I don't believe you will ever have the bug count drop to zero. I do believe in mathematical certainties though.
Should everybody drop C/C++/whatever and rush into the Rust train because Rust people has conjecture?
I ask the opposite question. What would happen if the only programming language left is C? Wouldn't we become better programmers and raise the bar so high that the bug count drops to 0?