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.

public/send.js

@@ -65,6 +65,12 @@ async function init() {
   const configRes = await fetch('/config');
   const { googleClientId } = await configRes.json();
 
+  // The GSI script tag is async/defer — without waiting for it to actually
+  // finish loading, this can run before `google` exists (a race that fails
+  // silently), which is why the button sometimes only appeared after a
+  // reload (once the script was already cached).
+  await window.googleGsiLoaded;
+
   google.accounts.id.initialize({
     client_id: googleClientId,
     callback: handleCredentialResponse,