Removing the borrow checker doesn’t break any correct programs — it just makes it so a huge amount of incorrect programs can be compiled. This is fine, since we mainly want to use mrustc to compile rustc, and we already know rustc can compile itself with no borrow checker errors.
In the very special case of proto-rust bootstrapping, the cost of not having borrow-checking can be paid back basically right away.
Not a user of Rust programs myself but am curious how users determine whether a Rust binary was compiled with mrustc or rustc.
Rust isn’t any kind of guarantee for end-users that some class of bugs doesn’t exist. It’s just a tool for programmers to make writing a certain class of bugs more difficult. If the programmer chooses to subvert the tool, they can.
(Also, I think most reasonable people would feel you were lying if you referred to a program that can’t be compiled by rustc as “written in rust”. Maybe “written in a dialect of rust” would be more accurate. Rust isn’t like C where there is a standard that multiple competing implementations target; the definition of rust is "what rustc accepts".)
This assumes the code is open source, you know which specific source code release was used and the author of the binary is making an effort to document the build environment that was used, e.g. through an attached SBOM.