Fix flaky regression test for Play->Back session persistence

The actual fix (main_menu.gd's _ready() checking ApiClient.has_token()
and restoring authenticated state via get_me()) was already correct --
verified this directly by instrumenting frame-by-frame state and seeing
it resolve properly in ~5 frames. The test wrapping it was the real
problem: _wait_for_menu_ready() only waited 2 frames total before
checking assertions, nowhere near enough for a real HTTP round-trip, so
every run failed on a fix that actually worked.

Root fix: added a real auth_state_ready signal to main_menu.gd, emitted
once _ready()'s session check has actually resolved (either branch --
authenticated or logged-out). The test now awaits that signal directly
instead of guessing from a frame count or from UI properties that are
ambiguous mid-await (AuthButtonsRow.visible is true both before the
check resolves AND in a genuine logged-out end state, which is exactly
what made an earlier polling-based fix attempt here also flaky).

Also fixed a second real bug in the test itself: unique_id used
Time.get_ticks_msec() (process-relative, not wall-clock), colliding
with a previous run's still-present data -- same flakiness class fixed
elsewhere this session, same fix (wall-clock ms + random suffix).

Verified: 12/12 passing across 3 consecutive runs against a real
backend.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

scripts/main_menu.gd

@@ -15,6 +15,14 @@ extends Control
 ## back to the login form.
 
 
+## Emitted once _ready()'s session check (has_token() + the awaited
+## get_me() call) has actually resolved, whichever way -- lets a test (or
+## anything else) know the UI state is trustworthy now, rather than
+## guessing from a frame count or from UI properties that are ambiguous
+## mid-await (e.g. AuthButtonsRow.visible is true both before the check
+## resolves AND in a genuine logged-out end state).
+signal auth_state_ready
+
 # ─── UI references ────────────────────────────────────────────────────────────
 
 var _auth_buttons_row: HBoxContainer
@@ -66,10 +74,12 @@ func _ready() -> void:
 			var username: String = str((me_result.data as Dictionary).get("username", ""))
 			if username != "":
 				_show_authenticated(username)
+				auth_state_ready.emit()
 				return
 
 	# Token missing or invalid — fall through to logged-out state.
 	_reset_to_logged_out()
+	auth_state_ready.emit()
 
 
 ## Restore the UI to the authenticated state (Play enabled, welcome label,