A Turing-complete language can, in theory and assuming no real-world limitations like RAM, do any computation. You can use Rust traits to compile Rust, or run a JVM instance, or whatever. Input and output are limited to what the compiler has access to, but you could perhaps have input as source files and output as text strings in a Rust binary.
That doesn't mean that the OP's hack can be used today, or even tomorrow for compiling-Rust-in-Rust. But you could, in theory, do so.
My point is that it's not actually going to be "you can type Rust code here and the trait system will magically compile it", it at best is going to be "you make some trait abomination that is somehow maps to the Rust program you wanted to compile and the machine will give you back some other abomination which you can through some encoding process get some sort of useful result out of".
