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/lib.rs

@@ -2,7 +2,6 @@
 
 use codex_arg0::Arg0DispatchPaths;
 use codex_cloud_requirements::cloud_requirements_loader;
-use codex_core::AuthManager;
 use codex_core::config::Config;
 use codex_core::config::ConfigBuilder;
 use codex_core::config_loader::CloudRequirementsLoader;
@@ -31,18 +30,19 @@ use crate::transport::auth::policy_from_settings;
 use crate::transport::route_outgoing_envelope;
 use crate::transport::start_stdio_connection;
 use crate::transport::start_websocket_acceptor;
+use codex_analytics::AppServerRpcTransport;
 use codex_app_server_protocol::ConfigLayerSource;
 use codex_app_server_protocol::ConfigWarningNotification;
 use codex_app_server_protocol::JSONRPCMessage;
 use codex_app_server_protocol::TextPosition as AppTextPosition;
 use codex_app_server_protocol::TextRange as AppTextRange;
-use codex_core::AppServerRpcTransport;
 use codex_core::ExecPolicyError;
 use codex_core::check_execpolicy_for_warnings;
 use codex_core::config_loader::ConfigLoadError;
 use codex_core::config_loader::TextRange as CoreTextRange;
 use codex_exec_server::EnvironmentManager;
 use codex_feedback::CodexFeedback;
+use codex_login::AuthManager;
 use codex_protocol::protocol::SessionSource;
 use codex_state::log_db;
 use tokio::sync::mpsc;