Earlier quoted context omitted.
Sure, thats one reason why IntoFuture and Future exist. Imo, in hindsight this is also main mistake in aysnc Rust: The whole async system should be build around IntoFuture rather than Future (async fn should return impl IntoFuture). That way you could pass around IntoFutures without being affected by auto traits leaking. Only when you actually call .await() or .poll() would the immovable Future materialize.
You're right that this is an important issue. I wrote a post explaining this in some detail, so at the very least we avoid having the same problem with generator functions: https://blog.yoshuawuyts.com/gen-auto-trait-problem
To dump some ideas on you: I think one missing piece might be that FnOnce() -> impl Future should implement IntoFuture. Async runtimes would then use IntoFuture in their APIs agressively.
I call this a "workload blueprint" at work. Its a closure/type that contains all the info to start the workload, but in a minimal form. In Rust terms this would be a buildprint that is ideally Send + Move + Forget + 'static, even if the actual work (and the backing struct of the Future) is !Send (e.g. It holds an Rc across await points).
Runtimes could use this for their advantage: There would be a global pool of "workload blueprint" that can be stolen by any executer thread, but once a !Send workload has started on one thread it can't be migrated to another.
This in combination with matklad's ideas about seperating TaskSend from ThreadSend (https://matklad.github.io/2023/12/10/nsfw.html) would solve most of my async pain points.