Summary

Fixes the Desktop↔Gateway config dual-write conflict described in #37751 by replacing the full-dict save_config() call with atomic per-key save_config_value() calls in the gateway's _handle_model_command --global path.

Root Cause

When /model <name> --global is used in the gateway, the old code:

1. Read the full config.yaml
2. Mutated the model dict in memory
3. Called save_config(cfg) which does a full YAML rewrite via atomic_yaml_write()

This full-dict rewrite can overwrite concurrent changes made by the Desktop UI (which uses regex field replacements), causing the contradictory config states described in #37751.

Fix

The CLI already avoids this problem by using save_config_value() (at cli.py:7802-7805) which calls atomic_roundtrip_yaml_update() — this updates only the requested dotted keys while preserving any other changes in the file.

This fix aligns the gateway with the same safe pattern:

• save_config_value("model.default", result.new_model)
• save_config_value("model.provider", result.target_provider) (only if provider changed)
• save_config_value("model.base_url", result.base_url) (only if base URL changed)

The scalar model coercion (model: <name> → model: {default: ...}) is handled internally by atomic_roundtrip_yaml_update(), so the explicit YAML parsing and dict manipulation code is no longer needed.

Testing

• Verified the gateway /model <name> --global command path uses save_config_value instead of save_config
• save_config_value is already tested in tests/cli/test_cli_save_config_value.py
• CLI /model --global behavior is unchanged (still uses save_config_value)
• Gateway picker flow (non-global session override) is unchanged

Closes #37751