feat: AI Router per session; Settings at the sidebar's foot; shipped tool icons

AI Router is a session's choice, not a workspace's:
- an AI Router select beside the model picker in the workspace home and a
  pane's run form (RouterPicker.tsx; hidden off Claude/Codex or with only
  Native; last pick remembered in agenthub.session.router)
- `router` on POST /api/lanes/:id/start and POST /api/run; Claude gets
  --settings=<data dir>/router-settings/<name>.json (0600, outranks project
  and user settings, key never in argv), Codex -p <name> + key via tmux -e
- dashboard_runs.router; a resume naming none goes back to it
- profile PATCH rewrites the session files, DELETE removes them
- gone: POST /api/lanes/:id/router/apply, LaneRouterRow, the Add-workspace
  router radio, the claude-/codex-home `routers` answers; clearLaneRouters
  takes the old keys out of each workspace's settings.local.json at start

Sidebar: Settings pinned to the foot, the collapse toggle beside it (stacked
on the rail) instead of in the brand's corner.

Icons: shipped SVG marks for claude, codex, openclaw and hermes (a
black/white pair swapped by theme) in client/public/ai-tools/, used by
AiToolIcon before the server's favicon.

server/routes/settings.js

@@ -25,7 +25,6 @@ const APP_VERSION = (() => {
 const { getSettingsPath, getClaudeHome, setClaudeHome } = require("../lib/claude-home");
 const { getCodexHome, setCodexHome } = require("../lib/codex-home");
 const { envOwned } = require("../lib/settings-env");
-const { reapplyCodexLayers, lanesWithMissingRouter } = require("../lib/router-apply");
 const pricingSync = require("../lib/pricing-sync");
 
 function getDbSize() {
@@ -285,7 +284,7 @@ router.get("/export", (_req, res) => {
 // overridden again at the next start), PUT validates, applies and persists
 // (server/lib/settings-env.js). EBADHOME = too broad, INVALID_PATH = any other
 // bad path, 500 ESAVE = the settings file could not be written.
-function homeRoutes(route, key, field, get, set, after = () => ({})) {
+function homeRoutes(route, key, field, get, set) {
   router.get(route, (_req, res) => {
     res.json({ [field]: get(), env_owned: envOwned.has(key) });
   });
@@ -308,20 +307,15 @@ function homeRoutes(route, key, field, get, set, after = () => ({})) {
         error: { code: bad ? err.code : "ESAVE", message: err.message },
       });
     }
-    res.json({ ok: true, [field]: resolved, env_owned: envOwned.has(key), ...after() });
+    res.json({ ok: true, [field]: resolved, env_owned: envOwned.has(key) });
   });
 }
 
-// GET/PUT /api/settings/claude-home — a PUT names the lanes whose AI Router
-// profile the new home's store does not have (they would start on native).
-homeRoutes("/claude-home", "CLAUDE_HOME", "claude_home", getClaudeHome, setClaudeHome, () => ({
-  routers: { missing: lanesWithMissingRouter() },
-}));
-// GET/PUT /api/settings/codex-home — a PUT re-writes every codex lane's AI
-// Router layer into the new home, or `codex -p <name>` fails at the next start.
-homeRoutes("/codex-home", "CODEX_HOME", "codex_home", getCodexHome, setCodexHome, () => ({
-  routers: reapplyCodexLayers(),
-}));
+// GET/PUT /api/settings/claude-home
+homeRoutes("/claude-home", "CLAUDE_HOME", "claude_home", getClaudeHome, setClaudeHome);
+// GET/PUT /api/settings/codex-home — a session's AI Router layer is written
+// into the current home at its spawn, so a move needs no repair here.
+homeRoutes("/codex-home", "CODEX_HOME", "codex_home", getCodexHome, setCodexHome);
 
 // POST /api/settings/cleanup — abandon stale sessions, purge old data
 router.post("/cleanup", (req, res) => {