feat(profile): public profile page and the switch in settings

/u/<name> shows a public profile without login. Settings get the switch and
the link; every settings save now sends the other saved values, and removing
the display name also makes the profile private.

src/App.jsx

@@ -11,6 +11,7 @@ import SettingsPage from './pages/SettingsPage.jsx';
 import GameLibrary from './pages/GameLibrary.jsx';
 import GameDetail from './pages/GameDetail.jsx';
 import WishlistPage from './pages/WishlistPage.jsx';
+import PublicProfilePage from './pages/PublicProfilePage.jsx';
 import AppHeader from './components/AppHeader.jsx';
 import {Toaster} from '@/components/ui/sonner';
 import {TooltipProvider} from '@/components/ui/tooltip';
@@ -49,6 +50,7 @@ function App() {
                         <Route path="/library" element={<GameLibrary/>}/>
                         <Route path="/library/:appid" element={<GameDetail/>}/>
                         <Route path="/wishlist" element={<WishlistPage/>}/>
+                        <Route path="/u/:name" element={<PublicProfilePage/>}/>
                         <Route path="/settings" element={<SettingsPage/>}/>
                         <Route path="/impressum" element={<ImprintPage/>}/>
                     </Routes>