👐 fix: Open/Close Sidebar Button Animation UX Regression from #10521 (#10694)

Problem:
--------
Commit 5ed1f2991 introduced a layout shift regression when opening the
sidebar. The UI would visibly "jump" as elements shifted right before
the animation completed. Closing the sidebar worked correctly.

Root Cause Analysis:
--------------------
The accessibility PR added a redundant `{navVisible && ...}` conditional
wrapper around the `<nav>` content inside Nav.tsx's `motion.div`. This
caused a race condition:

1. User clicks "Open Sidebar" button
2. `navVisible` state becomes `true`
3. React renders the `motion.div` AND its children simultaneously
4. The inner `{navVisible && (<nav>...)}` renders content at full width
   (320px/260px) BEFORE framer-motion applies `initial={{ width: 0 }}`
5. Brief flash of full-width content causes visible layout shift
6. Animation then starts from width: 0, but damage is done

The ref-based focus management (passing `openSidebarRef`/`closeSidebarRef`
through context) was suspected but was not the actual cause. However,
`requestAnimationFrame` focus calls during animation start could trigger
forced layout calculations, exacerbating the issue.

Solution:
---------
1. Remove redundant conditional rendering in Nav.tsx
   - The outer `{navVisible && (<motion.div>...)}` already controls
     visibility
   - The `overflow-x-hidden` class on motion.div clips content during
     animation
   - Content should always exist inside motion.div for smooth clipping

2. Replace ref-based focus with ID-based focus management
   - Refs passed through component tree can affect React's reconciliation
   - Using `document.getElementById()` decouples focus from render cycle
   - Exported `CLOSE_SIDEBAR_ID` and `OPEN_SIDEBAR_ID` constants for
     consistency

3. Delay focus until after animation completes
   - Changed from `requestAnimationFrame` to `setTimeout(..., 250)`
   - Animation duration is 200ms; 250ms ensures completion
   - Prevents layout thrashing during animation

4. Clean up prop drilling
   - Removed `openSidebarRef`/`closeSidebarRef` from Root.tsx context
   - Simplified Nav.tsx, Header.tsx, NewChat.tsx prop signatures
   - Updated ContextType to remove ref properties

Files Changed:
--------------
- client/src/routes/Root.tsx
- client/src/components/Nav/Nav.tsx
- client/src/components/Nav/NewChat.tsx
- client/src/components/Chat/Header.tsx
- client/src/components/Chat/Menus/OpenSidebar.tsx
- client/src/common/types.ts

Accessibility Note:
-------------------
The original inner conditional was added to prevent keyboard navigation
to hidden sidebar content for screen readers. This is still handled by:
- AnimatePresence unmounting the motion.div after exit animation
- The motion.div having width: 0 during exit (content not reachable)
- Screen readers typically skip content being animated out
- Other: removed non-existant prop from BookmarkNav

Testing:
--------
- Verified smooth animation when opening sidebar (no layout shift)
- Verified smooth animation when closing sidebar (unchanged)
- Verified focus transfers correctly between open/close buttons
- Verified keyboard navigation works as expected

client/src/routes/Root.tsx

@@ -1,4 +1,4 @@
-import React, { useState, useEffect, useRef } from 'react';
+import React, { useState, useEffect } from 'react';
 import { Outlet } from 'react-router-dom';
 import type { ContextType } from '~/common';
 import {
@@ -28,8 +28,6 @@ export default function Root() {
     const savedNavVisible = localStorage.getItem('navVisible');
     return savedNavVisible !== null ? JSON.parse(savedNavVisible) : true;
   });
-  const openSidebarRef = useRef<HTMLButtonElement>(null);
-  const closeSidebarRef = useRef<HTMLButtonElement>(null);
 
   const { isAuthenticated, logout } = useAuthContext();
 
@@ -75,24 +73,10 @@ export default function Root() {
               <Banner onHeightChange={setBannerHeight} />
               <div className="flex" style={{ height: `calc(100dvh - ${bannerHeight}px)` }}>
                 <div className="relative z-0 flex h-full w-full overflow-hidden">
-                  <Nav
-                    navVisible={navVisible}
-                    setNavVisible={setNavVisible}
-                    openSidebarRef={openSidebarRef}
-                    closeSidebarRef={closeSidebarRef}
-                  />
+                  <Nav navVisible={navVisible} setNavVisible={setNavVisible} />
                   <div className="relative flex h-full max-w-full flex-1 flex-col overflow-hidden">
                     <MobileNav navVisible={navVisible} setNavVisible={setNavVisible} />
-                    <Outlet
-                      context={
-                        {
-                          navVisible,
-                          setNavVisible,
-                          openSidebarRef,
-                          closeSidebarRef,
-                        } satisfies ContextType
-                      }
-                    />
+                    <Outlet context={{ navVisible, setNavVisible } satisfies ContextType} />
                   </div>
                 </div>
               </div>