It absolutely would not eliminate CVE-2014-3570 or many other of the cryptographic bugs that show up in OpenSSL and other TLS implementations.
This kind of axiomatic thinking that memory safety equals all safety almost resulted in Rust being less safe (IMO) than C in these other respects: the normal types in rust are all defined to overflow even though overflow is almost always a bug, and so you couldn't use static analysis on arbitrary rust codebases to warn you about these issues (where you can in C because signed overflow is undefined, so an analysis tool can always flag that as an error without it being a false positive). Fortunately, it sounds like Rust is moving to fix this ( http://discuss.rust-lang.org/t/a-tale-of-twos-complement/106... ), which is a big improvement.
But it's important to not be fuzzy about the risks. There is a well established link between safety equipment and risky behavior in driving and bicycling; (http://en.wikipedia.org/wiki/Risk_compensation). I don't mean to suggest that better isn't better and we shouldn't use safer tools, but if our response to safer tools is to lower our vigilance, even the slightest, we may get worse results.