Rust has concurrency issues for sure. Deadlocks are still a problem, as is lock poisoning, and sometimes dealing with the borrow checker in async/await contexts is very troublesome. Rust is great at many things, but safe Rust only eliminates certain classes of bugs, not all of them.
Regarding green threads: Rust originally started with them, but there were many issues. Graydon (the original author) has "grudgingly accepted" that async/await might work better for a language like Rust[1] in the end.
In any case, I think green threads and async/await are completely orthogonal to data race safety. You can have data race safety with green threeads (Rust was trying to have data-race safety even in its early green-thread era, as far as I know), and you can also fail to have data race-safety with async/await (C# might have fewer data-race safety footguns than Go but it's still generally unsafe).
in .NET, async/await does not protect you from data races and you are exposed to them as much as you are in Go, but there is a critical difference in that data races in .NET can never result (not counting unsafe) in memory safety violations. They can and will in Go.
Regarding green threads: Rust originally started with them, but there were many issues. Graydon (the original author) has "grudgingly accepted" that async/await might work better for a language like Rust[1] in the end.
In any case, I think green threads and async/await are completely orthogonal to data race safety. You can have data race safety with green threeads (Rust was trying to have data-race safety even in its early green-thread era, as far as I know), and you can also fail to have data race-safety with async/await (C# might have fewer data-race safety footguns than Go but it's still generally unsafe).
[1] https://graydon2.dreamwidth.org/307291.html