1. With Rust, you may lower the exposure, but the same classes of bug still remain. And of course, all the other non memory related bugs.
2. With C you may, if you wish, develop a big sensibility to race conditions, and stay alert. In general it is possible that C programmers have their "bugs antenna" a bit more developed than other folks.
3. With Rust, to decrease the "unsafe" sections amount, you need often to build abstractions that may be a bit unnatural.
4. Rust may create a false sense of security, and in the unsafe sections the programmer sometimes, when reviewing the code, is falsely convinced by the mandatory SAFETY comment. Like in the Linux Kernel bug, where such comment was hallucinated by a human that sometimes (not sure in this specific case, it's just an example) may be less used to do the "race spotting" process that C learns you to do.
5. With Rust, in case of a bug, the fix could no longer be the one-liner usually you see in C fixes, and can make the exposure time window larger. Sometimes fixing things in Rust means refactoring in non trivial ways.
6. With C, if there was the same amount of effort in creating wrappers to make kernel programming safer at the cost of other things, the surface of attack could also be lowered in a significant way (see for instance Redis use of sds.c: how many direct strings / pointers manipulation we avoid? The same for other stuff of course). Basically things like sds.c let you put a big part of the unsafe business in a self contained library.
So, is Rust an interesting language for certain features it has? Yes. Is Rust a silver bullet? No. So should Rust be "pushed" to others, hell no, and I suggest you to reply in the most firm way to people stressing you out to adopt Rust at all the costs.