Improve UX with merged run view, method badges, grouped encounters, and mobile nav

Merges the run dashboard into the encounters page as a unified view at /runs/:runId,
adds encounter method grouping in the modal and badges on route rows, improves the
home page with recent runs, adds mobile hamburger nav, and polishes empty states.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

frontend/src/types/game.ts

@@ -15,6 +15,7 @@ export interface Route {
   gameId: number
   order: number
   parentRouteId: number | null
+  encounterMethods: string[]
 }
 
 export interface RouteWithChildren extends Route {