refactor: simplify large functions & improve code-search clarity (follow-up to #2)

[Hrt-Htk]

Follow-up to #2 / #5. The module split landed the structural separation (one place for router lifecycle, no cross-file duplication, concern-named files), but two of #2's original goals were deliberately deferred to keep that pass mechanical and behaviour-preserving:

• functions are still large (no <40-line target met)
• proxy.py is still ~720 lines (the <100-line entry-point target was not met)

As discussed, the current code separates concerns by file but is still only so-so for code search / readability — the big handlers are monoliths, so grepping for a behaviour lands you in the middle of a 240-line function rather than a focused, well-named unit.

Goals

Improve simplification and code-search clarity without behaviour changes (same mechanical discipline as #2: byte-identical logic, verified by AST diff + live smoke test).

Tasks

• Split proxy_request (~240 lines in proxy.py) into named, greppable helpers — e.g. dead-worker retry, upstream forward, streaming-response wiring, request/response logging.
• Split ChatRouterManager.recover_worker (~190 lines in router_manager.py) into focused steps (probe / cycle / reload / verify).
• Slim the proxy.py entry point toward <100 lines by moving chat handlers (proxy_request, embed_forward, models_handler, props_handler) and preset generation (pick_setup, _model_preset_section, write_preset) into dedicated modules.
• router_http free-function dedup — extract the repeated auth-header + POST/poll boilerplate in _status / unload / _load_locked into shared helpers (the one Phase-2 item intentionally skipped; care needed to preserve ChatRouterManager's exact _loaded_at timing).
• Code-search clarity — consistent section markers, function-level docstrings on the new smaller units, and a short module-map (or expand CLAUDE.md's) so "where does X live" is answerable by grep/symbol search.

Acceptance

• No function >40 lines (best effort), proxy.py entry point <100 lines.
• No behaviour changes (AST diff of moved logic + live chat/embed smoke test, as in refactor: split monolithic proxy.py and reduce duplication with embed_proxy.py #2).
• Both proxies still compile, imports resolve, and serve real traffic.