It seems like you're trying to fix a social problem (programmers don't care about doing a good job) with a technical solution (change the programming languages). This simply doesn't work.
People who write C code ignoring warnings are the same people who in Rust will resort to writing unsafe with raw pointers as soon as they hit the first borrow check error. If you can't force them to care about C warnings, how are you going to force them to care about Rust safety?
I've seen this happen; it's not seen at large because the vast majority of people writing Rust code in public do it because they want to, not because they're forced.
I think it works, and quite well even. Defaults matter, a lot, and Rust and its stdlib do a phenomenal job at choosing really good ones, compared to many other languages. Cargo's defaults maybe not so much, but oh well.
In C, sloppy programmers will generally create crashy and insecure code, which can then be fixed and hardened later.
In Rust, sloppy programmers will generally create slower and bloated code, which can then be optimized and minimized later. That's still bad, but for many people it seems like a better trade-off for a starting point.
Inexperienced people who don't know better will make safe, bloated code in Rust.
Experienced people who simply ignore C warnings because they're "confident they know better" (as the other poster said) will write unsafe Rust code regardless of all the care in the world put in choosing sensible defaults or adding a borrow checker to the language. They will use `unsafe` and call it a day -- I've seen it happen more than once.
To fix this you have to change the process being used to write software -- you need to make sure people can't simply (for example) ignore C warnings or use Rust's `unsafe` at will.
And how many C programmers ignore such warnings because they are confident they know better?