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/cloud-tasks/src/lib.rs

@@ -12,6 +12,7 @@ use chrono::Utc;
 use codex_cloud_tasks_client::TaskStatus;
 use codex_git_utils::current_branch_name;
 use codex_git_utils::default_branch_name;
+use codex_login::default_client::get_codex_user_agent;
 use owo_colors::OwoColorize;
 use owo_colors::Stream;
 use std::cmp::Ordering;
@@ -58,7 +59,7 @@ async fn init_backend(user_agent_suffix: &str) -> anyhow::Result<BackendContext>
         });
     }
 
-    let ua = codex_core::default_client::get_codex_user_agent();
+    let ua = get_codex_user_agent();
     let mut http = codex_cloud_tasks_client::HttpClient::new(base_url.clone())?.with_user_agent(ua);
     let style = if base_url.contains("/backend-api") {
         "wham"
@@ -804,7 +805,7 @@ pub async fn run_main(cli: Cli, _codex_linux_sandbox_exe: Option<PathBuf>) -> an
     append_error_log(format!(
         "startup: wham_force_internal={} ua={}",
         force_internal,
-        codex_core::default_client::get_codex_user_agent()
+        get_codex_user_agent()
     ));
     // Non-blocking initial load so the in-box spinner can animate
     app.status = "Loading tasks…".to_string();

codex-rs/cloud-tasks/src/util.rs

@@ -8,7 +8,7 @@ use codex_core::config::Config;
 use codex_login::AuthManager;
 
 pub fn set_user_agent_suffix(suffix: &str) {
-    if let Ok(mut guard) = codex_core::default_client::USER_AGENT_SUFFIX.lock() {
+    if let Ok(mut guard) = codex_login::default_client::USER_AGENT_SUFFIX.lock() {
         guard.replace(suffix.to_string());
     }
 }
@@ -78,7 +78,7 @@ pub async fn build_chatgpt_headers() -> HeaderMap {
     use reqwest::header::USER_AGENT;
 
     set_user_agent_suffix("codex_cloud_tasks_tui");
-    let ua = codex_core::default_client::get_codex_user_agent();
+    let ua = codex_login::default_client::get_codex_user_agent();
     let mut headers = HeaderMap::new();
     headers.insert(
         USER_AGENT,