feat(live): paginate catalog and search server-side (#49)

Load Live channels from a filtered SQLite cursor API, keep navigation and guide consumers working across pages, and retain the QA reproduction coverage.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

public/js/app.js

@@ -130,6 +130,7 @@ class App {
 
         // Search clear buttons (global handler for all)
         document.querySelectorAll('.search-clear').forEach(btn => {
+            if (btn.parentElement.querySelector('#channel-search')) return;
             btn.addEventListener('click', () => {
                 const wrapper = btn.closest('.search-wrapper');
                 const input = wrapper?.querySelector('.search-input');