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/pages/PublicProfilePage.test.jsx

@@ -0,0 +1,61 @@
+// @vitest-environment jsdom
+import {afterEach, describe, expect, it, vi} from 'vitest';
+import {cleanup, render, screen, waitFor} from '@testing-library/react';
+import {MemoryRouter, Route, Routes} from 'react-router-dom';
+import PublicProfilePage from './PublicProfilePage.jsx';
+
+vi.mock('react-i18next', () => import('../test/i18n-mock.js'));
+
+const library = {
+    totalGames: 233, totalPlaytimeMinutes: 60000,
+    byStatus: {BACKLOG: 142, PLAYING: 6, FINISHED: 51, COMPLETED: 9, DROPPED: 25},
+    playing: [{appid: 220, name: 'Half-Life 2', playtimeMinutes: 300, status: 'PLAYING', rating: null}],
+    finished: [{appid: 620, name: 'Portal 2', playtimeMinutes: 900, status: 'COMPLETED', rating: 5}],
+};
+
+function mockBackends({profileStatus = 200, libraryStatus = 200} = {}) {
+    globalThis.fetch = vi.fn(async (url) => {
+        if (url.includes('/api/public/profiles/library')) {
+            return {ok: libraryStatus === 200, status: libraryStatus, json: async () => library};
+        }
+        return {ok: profileStatus === 200, status: profileStatus,
+            json: async () => ({displayName: 'Lobo', memberSince: '2026-03-02', ticket: 'the-ticket'})};
+    });
+}
+
+const renderPage = (name = 'Lobo') => render(
+    <MemoryRouter initialEntries={[`/u/${name}`]}>
+        <Routes><Route path="/u/:name" element={<PublicProfilePage/>}/></Routes>
+    </MemoryRouter>,
+);
+
+describe('PublicProfilePage', () => {
+    afterEach(cleanup);
+
+    it('asks Auth for the ticket, then Core for the library with it', async () => {
+        mockBackends();
+        renderPage('Lo bo');
+
+        expect(await screen.findByText('Portal 2')).toBeTruthy();
+        expect(fetch.mock.calls[0][0]).toMatch(/\/api\/public\/profiles\/Lo%20bo$/);
+        expect(fetch.mock.calls[1][1].headers['X-Profile-Ticket']).toBe('the-ticket');
+        expect(screen.getByText('Lobo')).toBeTruthy();
+        expect(screen.getByText('Half-Life 2')).toBeTruthy();
+        expect(screen.getByText('233')).toBeTruthy();
+    });
+
+    it('says the profile is unknown or private on a 404', async () => {
+        mockBackends({profileStatus: 404});
+        renderPage();
+
+        expect(await screen.findByText('PROFILE.NOT_FOUND')).toBeTruthy();
+        expect(fetch).toHaveBeenCalledTimes(1);
+    });
+
+    it('shows an error when Core fails', async () => {
+        mockBackends({libraryStatus: 500});
+        renderPage();
+
+        await waitFor(() => expect(screen.getByText('PROFILE.ERROR_LOAD')).toBeTruthy());
+    });
+});