fix: prevent Codex tab crash

[oz-for-oss]

Closes #10189

Description

• Resolve the preferred Codex model before opening the new Codex tab, close the modal before launch, and show a toast if Codex is unavailable.
• Scope the Codex model override to the newly-created terminal instead of mutating the shared default execution profile.
• Prevent Codex CLI-agent status updates from auto-opening rich input when the agent does not support rich status, with regression coverage.
• Mark the default-profile-id helper as test-only now that production code no longer calls it.

Linked Issue

• The linked issue is labeled ready-to-spec or ready-to-implement.

Screenshots / Videos

Not applicable; this is a launch-path crash fix.

Testing

• cargo fmt --all -- --check
• CARGO_BUILD_JOBS=1 cargo clippy --workspace --exclude warp_completer --all-targets --tests -- -D warnings
• CARGO_BUILD_JOBS=1 cargo clippy -p warp_completer --all-targets --tests -- -D warnings
• Attempted CARGO_BUILD_JOBS=1 cargo test -p warp ai::execution_profiles::profiles::tests -- --nocapture; the sandbox killed the warp lib test compile with SIGKILL before tests ran.

Agent Mode

• Warp Agent Mode - This PR was created via Warp's AI Agent Mode

CHANGELOG-BUG-FIX: Fixed a crash when opening a new Codex tab by avoiding mutation of the shared default execution profile and preventing unsupported Codex status updates from toggling rich input.