Not sure how much we could remove from Rust while keeping the problem tractable. The borrow checker is needed for compile time memory safety sans garbage collection. For borrow checking to be tractable, one needs shared references, exclusive references, owned values, the possibility for reference counting, and the possibility for interior mutability. This mandates smart pointers, that pretty much mandate generics. These various abstractions also mandate a capable standard library, unless the language would hardcode all these abstractions, which would endanger the low-levelness of the language (since you wouldn't be able to implement your own abstractions for e.g. embedded contexts)

For the demarcation between unsafe Rust and safe Rust to work, one needs encapsulation, so field privacy.

For concurrency, the language requires a way to mark types as thread safe, which requires a way to say things about structures, so a least a weak version of traits.

A smaller version of Rust probably exists, but I don't think we could remove as much of the language as we could imagine at first without compromising safety. It also wouldn't be "just a safe C", because C itself has heaps of accidental complexity (e.g. the way function pointers are declared, integer promotions, implicit conversions, the antiquated textual inclusion compilation model that is prone to ODR violations and complicates build systems, array decay, and so on...) that would need to be removed.

Hmmmmm, if you are not being forced by others into any particular Rust feature set (standard) and you are not being forced into any preexisting Rust codebase, then perhaps just forget the entire standard library, traits, async, etc. and go Bare Metal Rust[1], using only the Rust language features that you need and interfacing with external C APIs through Foreign Function Interface[2].

[1] https://google.github.io/comprehensive-rust/bare-metal.html

[2] https://doc.rust-lang.org/nomicon/ffi.html