feat(profile)!: replace sidebar and add avatar navigation

Rename the global Sidebar route, UI, assets, analytics, and payment return context to Profile. Add accessible message-avatar navigation and preserve the source character across auth and logout flows.

BREAKING CHANGE: /sidebar has been removed; use /profile instead.

e2e/specs/mock/auth/email-login-from-other-options.spec.ts

@@ -31,5 +31,5 @@ test("user can email login from other sign-in options and return to chat", async
   expect(["desktop", "android"]).toContain(loginRequest?.postDataJSON().platform);
 
   await expect(page.getByRole("textbox", { name: "Message" })).toBeVisible();
-  await expect(page.getByRole("button", { name: "Menu" })).toBeVisible();
+  await expect(page.getByRole("button", { name: "Profile" })).toBeVisible();
 });

e2e/specs/mock/auth/logout-from-profile.spec.ts

@@ -3,7 +3,7 @@ import { expect, test } from "@playwright/test";
 import { mockCoreApis } from "@e2e/fixtures/api-mocks";
 import {
   clearBrowserState,
-  defaultCharacterSidebarPath,
+  defaultCharacterProfilePath,
   defaultCharacterSplashPath,
   defaultCharacterChatUrl,
   enterChatFromSplash,
@@ -17,7 +17,7 @@ test.beforeEach(async ({ baseURL, context, page }) => {
   await mockCoreApis(page);
 });
 
-test("user can log out from the sidebar after email login", async ({
+test("user can log out from the profile after email login", async ({
   page,
 }) => {
   await enterChatFromSplash(page);
@@ -27,11 +27,11 @@ test("user can log out from the sidebar after email login", async ({
   await expect(page).toHaveURL(/\/auth(?:\?.*)?$/);
   await switchToEmailSignIn(page);
   await submitEmailLogin(page, { expectedUrl: defaultCharacterChatUrl });
-  await expect(page.getByRole("button", { name: "Menu" })).toBeVisible();
+  await expect(page.getByRole("button", { name: "Profile" })).toBeVisible();
 
-  await page.getByRole("button", { name: "Menu" }).click();
+  await page.getByRole("button", { name: "Profile" }).click();
   await expect(page).toHaveURL(
-    new RegExp(defaultCharacterSidebarPath.replace("?", "\\?") + "$"),
+    new RegExp(defaultCharacterProfilePath.replace("?", "\\?") + "$"),
   );
 
   const logoutRequestPromise = page.waitForRequest("**/api/auth/logout");