Rust has affine types, not linear. It also doesn't have mutable value semantics, it uses references, lifetimes, and borrowing.
To me it just seems like Rust has Linear types, and the compiler just inserts some code to destroy your values for you if you don't do it yourself.
I guess the only difference is that linear types can _force_ you to manually consume a value (not necessarily via drop)? Is that what you are going for?
See https://faultlore.com/blah/linear-rust/ for a (now pretty old but still pretty relevant, I think) exploration into what linear types would mean for Rust.