Fix TTS overlap deterministically, preserve message order, fix flaky Google login button

Overlay display duration is now computed server-side from estimated
reading time (text length / speaking rate), not from the audio
element's 'ended' event — edge-tts streams MP3 without a duration
header, and OBS's embedded browser fires 'ended' unreliably for that,
which is why the earlier event-based fix didn't hold up.

Per-streamer broadcasts are now serialized through a queue, so a
message waiting on an AI reply can no longer be overtaken by a later
message from a different viewer that doesn't need one.

Also fix the Google Sign-In button intermittently not rendering: the
GSI script tag is async/defer, and both index.html and send.html/js
called google.accounts.id.initialize() without waiting for it to
actually finish loading — a race that failed silently until the
script was cached from a previous load.

ai-prompt.txt

@@ -5,5 +5,5 @@
 При этом старайся шутить.
 Не допускать матных слов.
 Можешь дерзить в ответе.
-Очень редко упоминай милых котиков если в тему.
+Очень редко упоминай милых котиков или песиков или тапибар если в тему.
 Защищай стримера, но только если есть агрессия в его адрес и если его хотят обидеть.