add dev-only profiling with pprof, runtime/trace, and operation timing

Wire up Go's standard profiling toolkit so it's automatically available
in dev builds and completely absent from production. The profiling
package uses build tags (dev/!dev) to eliminate all pprof, trace, and
timing code from release binaries with zero new dependencies.

- backend/profiling: pprof HTTP server on :6060, /debug/trace endpoint,
  block/mutex profiling, and TimeOp helper for structured operation timing
- scripts/profile.sh: interactive menu-driven script that auto-selects
  free ports (8080-8089) so multiple profiles can be open simultaneously
- Instrumented key operations: app init, database init, player load/restore,
  queue set/restore
- Makefile targets: profile, profile-cpu, profile-heap, profile-trace
- .gitignore: exclude trace-*.out and *.pprof artifacts

Makefile

@@ -36,3 +36,16 @@ install: ## Install all development dependencies (Go tools, frontend packages)
 
 setup: install ## Install dependencies and set up git hooks
 	go tool lefthook install
+
+# Profiling (dev builds only — pprof server on :6060 starts automatically)
+profile: ## Open interactive profiling menu (CPU, heap, trace, etc.)
+	@./scripts/profile.sh
+
+profile-cpu: ## Capture CPU profile and open flame graph in browser
+	@./scripts/profile.sh cpu
+
+profile-heap: ## Capture heap profile and open in browser
+	@./scripts/profile.sh heap
+
+profile-trace: ## Capture execution trace and open trace viewer
+	@./scripts/profile.sh trace