Deduplicate built-in provider default option

[euforic]

Summary

• collapse the built-in provider and default rows when built-in OAuth is the current default
• preserve the explicit built-in fallback when a named provider is the default
• normalize existing explicit built-in selections to the unified displayed row

Validation

• swift test (270 passed, 9 opt-in tests skipped, 0 failures)

---

^{Need help on this PR? Tag @codesmith-bot with what you need. Autofix is disabled.}

[Copilot]

Copilot was unable to review this pull request because the user who requested the review has reached their quota limit.

[chatgpt-codex-connector]

Codex Review Summary

This comment shows the latest Codex review activity on this pull request.

Review	Status	Commit	Review trigger
📝 Code Review	✅ Completed 2026-09-01T21:07:08.819425Z	ada63fb	PR opened

ℹ️ About Codex in GitHub

Your team has set up Codex to review pull requests in this repo. Reviews are triggered when you

• Open a pull request for review
• Mark a draft as ready
• Comment "@codex review" or "@codex security review".

Codex reacts with 👀 while any review is running, comments if it has suggestions, and reacts with 👍 once all reviews finish with no findings.

[chatgpt-codex-connector]

💡 Codex Review

Here are some automated review suggestions for this pull request.

Reviewed commit: ada63fb1f8

ℹ️ About Codex in GitHub

Your team has set up Codex to review pull requests in this repo. Reviews are triggered when you

• Open a pull request for review
• Mark a draft as ready
• Comment "@codex review".

If Codex has suggestions, it will comment; otherwise it will react with 👍.

Codex can also answer questions or update the PR. Try commenting "@codex address that feedback".

[chatgpt-codex-connector]

Update the documented provider option names

When no named default exists, the picker now exposes Built-in Codex (OAuth) — Default instead of the README's documented Use Default option, while README.md:86-90 still directs users to choose Use Default. Users following Quick Start therefore cannot find the named control; update the focused documentation with the new conditional labels in this change.

AGENTS.md reference: AGENTS.md:L50-L51

Useful? React with 👍 / 👎.