This makes combining Rust into already existing C code a hassle, as it is generally worse then even just trying to use C and C++ together which is already fairly annoying and error-prone. This is compounded by Rust's safety goals which generally require a different design to a problem then the "C-style" approach would be - IE. You're probably not going to get good Rust code by simply replacing a .c file with a Rust file, because the entire time you're going to have to make use of the unsafe C functions that your program includes.
Perhaps the bottom line is that (for good reason) Rust is much more then just an improved C. And because of that, combining C and Rust is always going to feel like combining two separate systems together with a compatibility layer in-between - because that's really what it is - and that's just not very attractive.
I know little about Zig (I'm reading about it for the first time now) and don't at all predict it to replace C. That said, I would love a language which has a focus on maintaining very good compatibility with C while fixing the various pieces of its design that are fixable within the bounds of the language. For example, I noticed it mentioned in the README that in Zig that pointers are nonnull by default, and can be made nullable by adding the 'maybe' attribute. This is a feature I would love in C, but really isn't there (gcc has a nonnull attribute, but it is really essentially useless and nothing like you'd want). Essentially, I would like a language where it feels like writing code for the same system, but in a different (better) syntax - where I can drop it into a project and it works together with the system like any other .c file (Though with a different compiler, obviously), but the code inside is much better then what you can do in standard C. I think that such a language could retain the reasons why people (like me) still like C, while fixing a lot of the uglier sides to the language that everybody is aware of but aren't going to get fixed any time soon. And perhaps most importantly, I think such a language is definitely possible (Though it may not be able to employ all the features you may want) - but I don't have the time nor probably the skills to really do it well besides list off the things it should fix and shouldn't fix.
As for starting a project though, if you ignore difficulty to learn then I would agree with you: There's probably little reason not to just go for Rust if you're already willing to learn a new language. For most projects the issues I outlined above don't really matter assuming the entire project is written in Rust - besides the want for a language closer to C in design. That said, I readily concede that I have no idea if Rust's borrow-checking semantics could ever work without the extra features they added that take the design away from being 'C-like' - and considering that's one of the definitive features of Rust, is easily worth losing the 'C-like' detail if it is necessary.
But when considering a project like the Linux Kernel, the GNU coreutils, git, GTK, and other various large projects where conversion to Rust is probably impossible without a complete rewrite, being able to use a 'better' C while still retaining the aspects that people like would probably be a nice step in the right direction. The chances of it happening are nil, but it would still be a step in the right direction.