curl: let's not rush, 23 years is not enough time for people to update their compilers.
Rust: the compiler released last Thursday is the oldest supported version.
It's for the proponents of rust.
Personally, I'm largely convinced that writing in rust will lead to safer implementations... now I'm just waiting for the full-featured, well-though-out, stable, useful rust implementations to appear.
I'm not married to curl. Where's "rurl" or whatever? If it exists and does something useful as well as curl (or better), I'll happily use it.
So there is really no reason to use it Rust, use smaller, protocol-specific libs, less security problems and if you need multi-protocol support just switch by protocol and maybe write some wrappers.
And for calling it from C, "Rewrite it in Rust but make ABI C compatible" means you have to repeat a lot of Curl's idiosynchracies all while recreating all of the functionality. Looking at https://curl.se/docs/security.html there isn't that many of them that Rust safety would prevent so the real gains from that aren't probably all that great.
Now OpenSSL on another hand...
curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | shAnd start a rewrite themselves afterwards.