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/Cargo.lock

@@ -1401,6 +1401,7 @@ dependencies = [
  "base64 0.22.1",
  "chrono",
  "clap",
+ "codex-analytics",
  "codex-app-server-protocol",
  "codex-arg0",
  "codex-backend-client",
@@ -1417,6 +1418,7 @@ dependencies = [
  "codex-otel",
  "codex-protocol",
  "codex-rmcp-client",
+ "codex-rollout",
  "codex-sandboxing",
  "codex-shell-command",
  "codex-state",
@@ -1573,7 +1575,7 @@ dependencies = [
  "anyhow",
  "codex-backend-openapi-models",
  "codex-client",
- "codex-core",
+ "codex-login",
  "codex-protocol",
  "pretty_assertions",
  "reqwest",
@@ -1642,6 +1644,7 @@ dependencies = [
  "codex-tui",
  "codex-utils-cargo-bin",
  "codex-utils-cli",
+ "codex-utils-path",
  "codex-windows-sandbox",
  "libc",
  "owo-colors",
@@ -1698,6 +1701,7 @@ dependencies = [
  "chrono",
  "codex-backend-client",
  "codex-core",
+ "codex-login",
  "codex-otel",
  "codex-protocol",
  "hmac",
@@ -1995,6 +1999,7 @@ dependencies = [
  "codex-core",
  "codex-feedback",
  "codex-git-utils",
+ "codex-login",
  "codex-otel",
  "codex-protocol",
  "codex-utils-absolute-path",
@@ -2301,6 +2306,7 @@ dependencies = [
  "codex-core",
  "codex-exec-server",
  "codex-features",
+ "codex-login",
  "codex-protocol",
  "codex-shell-command",
  "codex-utils-cli",
@@ -2696,6 +2702,7 @@ dependencies = [
  "codex-mcp",
  "codex-otel",
  "codex-protocol",
+ "codex-rollout",
  "codex-shell-command",
  "codex-state",
  "codex-terminal-detection",
@@ -3197,6 +3204,7 @@ dependencies = [
  "codex-core",
  "codex-exec-server",
  "codex-features",
+ "codex-login",
  "codex-protocol",
  "codex-utils-absolute-path",
  "codex-utils-cargo-bin",
@@ -6012,6 +6020,7 @@ version = "0.0.0"
 dependencies = [
  "anyhow",
  "codex-core",
+ "codex-login",
  "codex-mcp-server",
  "codex-terminal-detection",
  "codex-utils-cargo-bin",