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/config_api.rs

@@ -1,6 +1,7 @@
 use crate::error_code::INTERNAL_ERROR_CODE;
 use crate::error_code::INVALID_REQUEST_ERROR_CODE;
 use async_trait::async_trait;
+use codex_analytics::AnalyticsEventsClient;
 use codex_app_server_protocol::ConfigBatchWriteParams;
 use codex_app_server_protocol::ConfigReadParams;
 use codex_app_server_protocol::ConfigReadResponse;
@@ -16,7 +17,6 @@ use codex_app_server_protocol::NetworkDomainPermission;
 use codex_app_server_protocol::NetworkRequirements;
 use codex_app_server_protocol::NetworkUnixSocketPermission;
 use codex_app_server_protocol::SandboxMode;
-use codex_core::AnalyticsEventsClient;
 use codex_core::ThreadManager;
 use codex_core::config::Config;
 use codex_core::config::ConfigService;
@@ -510,15 +510,15 @@ fn config_write_error(code: ConfigWriteErrorCode, message: impl Into<String>) ->
 #[cfg(test)]
 mod tests {
     use super::*;
-    use codex_core::AnalyticsEventsClient;
-    use codex_core::AuthManager;
-    use codex_core::CodexAuth;
+    use codex_analytics::AnalyticsEventsClient;
     use codex_core::config_loader::NetworkDomainPermissionToml as CoreNetworkDomainPermissionToml;
     use codex_core::config_loader::NetworkDomainPermissionsToml as CoreNetworkDomainPermissionsToml;
     use codex_core::config_loader::NetworkRequirementsToml as CoreNetworkRequirementsToml;
     use codex_core::config_loader::NetworkUnixSocketPermissionToml as CoreNetworkUnixSocketPermissionToml;
     use codex_core::config_loader::NetworkUnixSocketPermissionsToml as CoreNetworkUnixSocketPermissionsToml;
     use codex_features::Feature;
+    use codex_login::AuthManager;
+    use codex_login::CodexAuth;
     use codex_protocol::protocol::AskForApproval as CoreAskForApproval;
     use pretty_assertions::assert_eq;
     use serde_json::json;