core: remove cross-crate re-exports from lib.rs (#16512)

## Why

`codex-core` was re-exporting APIs owned by sibling `codex-*` crates,
which made downstream crates depend on `codex-core` as a proxy module
instead of the actual owner crate.

Removing those forwards makes crate boundaries explicit and lets leaf
crates drop unnecessary `codex-core` dependencies. In this PR, this
reduces the dependency on `codex-core` to `codex-login` in the following
files:

```
codex-rs/backend-client/Cargo.toml
codex-rs/mcp-server/tests/common/Cargo.toml
```

## What

- Remove `codex-rs/core/src/lib.rs` re-exports for symbols owned by
`codex-login`, `codex-mcp`, `codex-rollout`, `codex-analytics`,
`codex-protocol`, `codex-shell-command`, `codex-sandboxing`,
`codex-tools`, and `codex-utils-path`.
- Delete the `default_client` forwarding shim in `codex-rs/core`.
- Update in-crate and downstream callsites to import directly from the
owning `codex-*` crate.
- Add direct Cargo dependencies where callsites now target the owner
crate, and remove `codex-core` from `codex-rs/backend-client`.

codex-rs/mcp-server/src/message_processor.rs

@@ -2,14 +2,14 @@ use std::collections::HashMap;
 use std::sync::Arc;
 
 use codex_arg0::Arg0DispatchPaths;
-use codex_core::AuthManager;
 use codex_core::ThreadManager;
 use codex_core::config::Config;
-use codex_core::default_client::USER_AGENT_SUFFIX;
-use codex_core::default_client::get_codex_user_agent;
 use codex_core::models_manager::collaboration_mode_presets::CollaborationModesConfig;
 use codex_exec_server::EnvironmentManager;
 use codex_features::Feature;
+use codex_login::AuthManager;
+use codex_login::default_client::USER_AGENT_SUFFIX;
+use codex_login::default_client::get_codex_user_agent;
 use codex_protocol::ThreadId;
 use codex_protocol::protocol::SessionSource;
 use codex_protocol::protocol::Submission;