meet-bot: implement POST /send_chat endpoint

[siddseethepalli]

Summary

• Replace 501 stub in bot HTTP server with real /send_chat handler
• Validate body via SendChatCommand Zod; enforce 2000-char limit
• Call sendChat helper from chat-bridge; return 200/400/401/502 as appropriate
• Unit tests cover auth, validation, success, Playwright failure paths

Part of plan: meet-phase-2-chat.md (PR 1 of 8)

[chatgpt-codex-connector]

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[chatgpt-codex-connector]

Serialize /send_chat requests before invoking onSendChat

/send_chat handlers run concurrently, but they all drive the same Meet chat input on a single Playwright page. If two authenticated requests overlap, one call can fill() over the other's text before press("Enter"), causing the wrong message to be sent (or sent twice) even though both HTTP requests may return success. Adding a per-server mutex/queue around onSendChat would make message delivery deterministic under parallel sends/retries.

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[siddseethepalli]

Follow-up shipped in #26265 — addresses consolidated review feedback (chat concurrency mutex, dispose teardown, wake adapter timeout/ghost prevention, tokenEstimationProvider forwarding, E2E test flakiness).