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/app-server/src/codex_message_processor.rs

@@ -20,6 +20,7 @@ use crate::thread_status::resolve_thread_status;
 use chrono::DateTime;
 use chrono::SecondsFormat;
 use chrono::Utc;
+use codex_analytics::AnalyticsEventsClient;
 use codex_app_server_protocol::Account;
 use codex_app_server_protocol::AccountLoginCompletedNotification;
 use codex_app_server_protocol::AccountUpdatedNotification;
@@ -180,9 +181,6 @@ use codex_arg0::Arg0DispatchPaths;
 use codex_backend_client::Client as BackendClient;
 use codex_chatgpt::connectors;
 use codex_cloud_requirements::cloud_requirements_loader;
-use codex_core::AnalyticsEventsClient;
-use codex_core::AuthManager;
-use codex_core::CodexAuth;
 use codex_core::CodexThread;
 use codex_core::Cursor as RolloutCursor;
 use codex_core::ForkSnapshot;
@@ -193,9 +191,6 @@ use codex_core::SteerInputError;
 use codex_core::ThreadConfigSnapshot;
 use codex_core::ThreadManager;
 use codex_core::ThreadSortKey as CoreThreadSortKey;
-use codex_core::auth::AuthMode as CoreAuthMode;
-use codex_core::auth::CLIENT_ID;
-use codex_core::auth::login_with_api_key;
 use codex_core::config::Config;
 use codex_core::config::ConfigOverrides;
 use codex_core::config::NetworkProxyAuditMetadata;
@@ -207,7 +202,6 @@ use codex_core::config_loader::CloudRequirementsLoadErrorCode;
 use codex_core::config_loader::CloudRequirementsLoader;
 use codex_core::config_loader::LoaderOverrides;
 use codex_core::config_loader::load_config_layers_state;
-use codex_core::default_client::set_default_client_residency_requirement;
 use codex_core::error::CodexErr;
 use codex_core::error::Result as CodexResult;
 use codex_core::exec::ExecCapturePolicy;
@@ -233,9 +227,6 @@ use codex_core::read_head_for_summary;
 use codex_core::read_session_meta_line;
 use codex_core::rollout_date_parts;
 use codex_core::sandboxing::SandboxPermissions;
-use codex_core::state_db::StateDbHandle;
-use codex_core::state_db::get_state_db;
-use codex_core::state_db::reconcile_rollout;
 use codex_core::windows_sandbox::WindowsSandboxLevelExt;
 use codex_core::windows_sandbox::WindowsSandboxSetupMode as CoreWindowsSandboxSetupMode;
 use codex_core::windows_sandbox::WindowsSandboxSetupRequest;
@@ -244,10 +235,16 @@ use codex_features::Feature;
 use codex_features::Stage;
 use codex_feedback::CodexFeedback;
 use codex_git_utils::git_diff_to_remote;
+use codex_login::AuthManager;
+use codex_login::AuthMode as CoreAuthMode;
+use codex_login::CLIENT_ID;
+use codex_login::CodexAuth;
 use codex_login::ServerOptions as LoginServerOptions;
 use codex_login::ShutdownHandle;
 use codex_login::auth::login_with_chatgpt_auth_tokens;
 use codex_login::complete_device_code_login;
+use codex_login::default_client::set_default_client_residency_requirement;
+use codex_login::login_with_api_key;
 use codex_login::request_device_code;
 use codex_login::run_login_server;
 use codex_mcp::mcp::auth::discover_supported_scopes;
@@ -284,6 +281,9 @@ use codex_protocol::protocol::W3cTraceContext;
 use codex_protocol::user_input::MAX_USER_INPUT_TEXT_CHARS;
 use codex_protocol::user_input::UserInput as CoreInputItem;
 use codex_rmcp_client::perform_oauth_login_return_url;
+use codex_rollout::state_db::StateDbHandle;
+use codex_rollout::state_db::get_state_db;
+use codex_rollout::state_db::reconcile_rollout;
 use codex_state::StateRuntime;
 use codex_state::ThreadMetadata;
 use codex_state::ThreadMetadataBuilder;