No, because the program may not be able to actually understand the syntax of your favorite programming language. As I mentioned in another comment, FRACTRAN is Turing-complete and you cannot just shove a string of Rust code at it because it has no idea what a string is; the best you can do is implement the "compiler" as working on some numerically-encoded version of Rust and producing some encoded version of a native binary on the other side. So you've lost the syntax because you've had to do that additional encoding.
But your computer does understand the distinction between a string numerically encoded as consecutive bytes in memory, versus a string encoded as a arbitrary-precision integer equal to ( 2^(first ASCII value) * 3^(second ASCII value) * 5^(third '') * 7^(fourth '') * ... ).
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".
