> Rust’s borrow checker doesn’t actually borrow anything either

Why would it? It's called the borrow checker, not the borrower. So it checks that your borrows are valid.

std::move looks and feels like a function, but it doesn't do what it says, it makes objects movable but does never moves them (that's up to whatever is using the value afterwards). If you want something similar in Rust, Pin is a much better candidate.