First things first: below Rust only opcodes shall remain. Even ASM (&DSL) shall find its place in Rust. Rust has to stick to bare metal, at all costs. This is paramount.
The proposal (v1) is not about a minimal runtime, but precisely about a zero-cost runtime.
A zero-cost runtime is in fact a compiler abstraction, which in this case projects a multi-threaded async/await program onto a single-threaded async/await execution system. For analogy, think std::unique_ptr.
This introduces no dependencies into Rust programs, just in the compiler (that is, one more module to write). It is what .Net did when got async/await in C(from F)#, and IIUC it is the same thing that @withoutboats & @kpreid propose (it is also the right idea).
It is a proposal intended to plug the first gap in the async/await Rust eco-system. De facto, there are two Rust versions now, out of which one only compiles on systems able to multi-thread. This is an already existing dependency in the language and should be avoided at all costs.
The second gap to be plugged is exactly the incompatible divergence between the async runtimes themselves (e.g. tokio vs. async-std vs...), which shall not be allowed to go beyond test-and-prove concepts. Each of these runtimes already creates its own eco-system (and on top of that each runtime kinda rewrites the entire std to its own image). A standardized mechanism of compatibility between these eco-systems/runtimes is necessary.
The gap(s) may appear small at this time, but there are precedents: D programming language knew the war between the two roses named "phobos" and "tango", which were competing for the status of the-one-true-standard-library, which plagued the project beyond immagination. It took D2, almost a new language, to stop that, and even that only because the "tango" team gave up. Similar fates are to be avoided in a (solid) Rusty world.
One more thing: the as of yet most under-appreciated quality of Rust is not its safety, but its explainability: it codes an intent unambiguously (parenthesis: the post about the alternative proposal of using unique/multiple references instead of mutable/immutable references saw that correctly towards its last paragraphs, and in a way acknowledged that mutable/immutable is the better choice in the end 1). Coding the intent is paramount for the future, when codebases will be gigantesque and all the (intelligent?) tools that will analyze (and, for sure, write) that code will rely essentially on this markup of the intent. It is this quality that makes Rust the first (and maybe the single-ever-needed) universal programming language (UPL): able to cover everything from the bare metal to humongous systems in an explainable way. C did not make it there not because of its lack of expressiveness, but precisely because of its lack of unambiguous intent/explainability. Ditto, C++. Other languages did not touch the bare metal, so "why bother?".
As of now, the areas still restricted to Rust remain the browser (because wasm is still purposefully restricted) and the scripting (getting a bit Rustic at times...). As for universality, the mutable value semantics of Val are the strongest competing idea, a paradigm that Rust itself might (have to) integrate at a future moment.