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/index.html

@@ -4,7 +4,10 @@
 <meta charset="UTF-8" />
 <title>Send2Streamer — сообщения от зрителей прямо на стрим</title>
 <meta name="viewport" content="width=device-width, initial-scale=1" />
-<script src="https://accounts.google.com/gsi/client" async defer></script>
+<script>
+  window.googleGsiLoaded = new Promise((resolve) => { window.resolveGoogleGsiLoaded = resolve; });
+</script>
+<script src="https://accounts.google.com/gsi/client" async defer onload="resolveGoogleGsiLoaded()"></script>
 <style>
   body {
     font-family: system-ui, sans-serif;
@@ -87,6 +90,12 @@
     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,