A lot of that, I suspect, is that Rust's borrow checker makes specifying ownership correctly for callback chains extremely painful and unergonomic.
In my experience, closure-based async programming is only sufficiently painless in managed languages to be worth using. When you're dealing with manual memory management (or, rather, smart pointer memory management), you spend a lot more time trying to make the captures hold onto stuff.
I'd agree with this, and emphasize the point that this stuff is really tricky to get right without GC. Fighting the borrow checker is somewhat expected when you're dealing with this level of inherent complexity in your memory management.
One of the key reasons for shipping async/await is that it erases almost all of this difficulty and lets you write straight-line code again.
In my experience, closure-based async programming is only sufficiently painless in managed languages to be worth using. When you're dealing with manual memory management (or, rather, smart pointer memory management), you spend a lot more time trying to make the captures hold onto stuff.