[codex] Remove async_trait from first-party code (#27475)

## Why

First-party async traits should expose their `Send` contracts explicitly
without requiring `async_trait`. This completes the migration pattern
established in #27303 and #27304.

## What changed

- Replaced the remaining first-party `async_trait` traits with native
return-position `impl Future + Send` where statically dispatched and
explicit boxed `Send` futures where object safety is required.
- Kept implementations behavior-preserving, outlining existing async
bodies into inherent methods where that keeps the diff reviewable.
- Removed all direct first-party `async-trait` dependencies and the
workspace dependency declaration.
- Added a cargo-deny policy that permits `async-trait` only through the
remaining transitive wrapper crates.
- Updated `rand` from 0.8.5 to 0.8.6 to resolve RUSTSEC-2026-0097 and
keep the full cargo-deny check passing.

## Validation

- `just test -p codex-exec-server`: 216 passed, 2 skipped.
- `just test -p codex-model-provider`: 39 passed.
- `just test -p codex-core` and `just test`: changed tests passed;
remaining failures are environment-sensitive suites unrelated to this
migration.
- `cargo deny check`
- `just fix`
- `just fmt`
- `cargo shear`
- `just bazel-lock-check`

codex-rs/tui/src/workspace_command.rs

@@ -138,9 +138,8 @@ pub(crate) trait WorkspaceCommandExecutor: Send + Sync {
     /// Runs a workspace command and returns captured output or an app-server request error.
     ///
     /// Callers should treat errors as infrastructure failures and should treat successful output
-    /// with a non-zero exit code as ordinary command failure. Returning a future instead of using
-    /// `async_trait` keeps the trait object-safe while matching the repo's native async trait
-    /// conventions.
+    /// with a non-zero exit code as ordinary command failure. Returning a boxed future keeps the
+    /// trait object-safe.
     fn run(
         &self,
         command: WorkspaceCommand,