feat(tiles): Tiles header button with a session picker, and a tile's + menu

A Tiles button beside Split in the header, opt-in through the per-device
showTileGridButton setting (read in applyHeaderVisibilitySettings; the
settings checkbox, displayKeys membership and schema exclusion follow with
persistence) and hard-gated like Split: hidden by its --hidden marker, a JS
width check with a live media listener, a @media (max-width: 1179px) backstop
and never in a solo window. While the grid is open the button closes it and
reads as pressed.

Closed, it opens a picker: a checkbox per open session in tab order (never one
popped out to its own window; one with no PTY is offered, its tile shows the
Attach overlay), names as text, preselected with the grid this tab last left,
else the active session and an open split's two. Boxes past what the window
can fit are disabled with the count shown, and Open opens the grid on the
checked sessions, focusing the active one if checked. Escape and an outside
click close it; its close method is idempotent and the global Escape handler
calls it.

A tile's + lists the open sessions not yet tiled; picking one adds it and
focuses it (a human selection). A grid that already holds what the window fits
disables the entries. "New session in this case" waits for auto-join.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

src/web/public/index.html

@@ -192,6 +192,7 @@
         <button class="btn-icon-header btn-file-viewer" onclick="app.toggleFileBrowserButton()" title="File Viewer" aria-label="Open file viewer" aria-expanded="false"><svg width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M3 7a2 2 0 0 1 2-2h4l2 2h8a2 2 0 0 1 2 2v8a2 2 0 0 1-2 2H5a2 2 0 0 1-2-2z"/></svg></button>
         <button class="btn-icon-header btn-multimonitor btn-multimonitor--hidden" onclick="app.launchMultiMonitor()" title="Open Codeman across all displays" aria-label="Open Codeman across all displays"><svg width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2" y="4" width="13" height="9" rx="1.5"/><rect x="11" y="9" width="11" height="8" rx="1.5"/></svg></button>
         <button class="btn-icon-header btn-split btn-split--hidden" onclick="app.openSplitPicker(event)" title="Split: open a second session beside this one" aria-label="Split: open a second session beside this one" aria-pressed="false"><svg width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2" y="3" width="20" height="18" rx="2"/><line x1="12" y1="3" x2="12" y2="21"/></svg></button>
+        <button class="btn-icon-header btn-tile-grid btn-tile-grid--hidden" onclick="app.openTilePicker(event)" title="Tiles: show several sessions side by side" aria-label="Tiles: show several sessions side by side" aria-pressed="false"><svg width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="3" y="3" width="8" height="8" rx="1"/><rect x="13" y="3" width="8" height="8" rx="1"/><rect x="3" y="13" width="8" height="8" rx="1"/><rect x="13" y="13" width="8" height="8" rx="1"/></svg></button>
         <button class="btn-icon-header btn-ultracode-agents btn-ultracode-agents--hidden" onclick="app.toggleUltracodeAgentsPanel()" title="Ultracode / Workflow agents" aria-label="Open ultracode workflow agents"><svg width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><circle cx="6" cy="6" r="2.5"/><circle cx="6" cy="18" r="2.5"/><circle cx="18" cy="12" r="2.5"/><path d="M8.2 7.2 15.6 11M8.2 16.8 15.6 13"/></svg></button>
         <div class="header-plan-usage header-plan-usage--hidden" id="planUsageChip" title="Claude and Codex plan usage limits">—</div>
         <button class="btn-icon-header btn-notifications" onclick="app.toggleNotifications()" title="Notifications" aria-label="Toggle notifications" style="display:none;">