https://www.cvedetails.com/vulnerabilities-by-types.php is a bit more clear. It's xss, SQL, then memory. The first two are not possible to enforce a fix on - you can always make a decision to do something bad with no visible annotation. Even then, rich types like in rust make safe interfaces easier to produce. But rust tackles the next class of issues - one that you can verify to be safe or require an explicit "unsafe" around it.
As for preventing XSS and SQL injections, that's what good web frameworks do. If your framework encourages you to write raw unescaped SQL, or doesn't provide sensible defaults around content policies, then no matter what language it's in, there are going to be issues (and maybe if we called these frameworks "unsafe" then we'd get somewhere with fixing them).
My feeling is in the specific instance of using rust in apt, this is most likely a good thing (though I hope existing well tested rust libraries are used rather than NIHing them and introducing new bugs), but so far Ubuntu's rustification has not gone smoothly, so I'm more wary of the changes that e.g. improvements to Firefox via rust.
I think that's much more likely to introduce bugs.
Think of it that way, a lot of the Rust libraries are rewriting existing copyleft libraries in permissive licenses, so they cannot look at the original code, dooming them to repeat the mistakes that were made in the original code and having to fix them all over again on their own (as both go from "oh this is simple" to "oh another corner case").
I just want to translate code 1:1 to Rust, reusing my existing knowledge, design decisions, and tests. It should behave _exactly_ the same as before, just memory safe.