[Elastic EP] Fix recovery lifecycle and add manual coverage (#31744)

Co-authored-by: Shangming Cai <csmthu@gmail.com>

python/sglang/srt/managers/scheduler.py

@@ -885,6 +885,12 @@ class Scheduler(
         if model_runner.token_to_kv_pool.post_capture_active:
             model_runner.post_capture_resize_kv_pool()
 
+        if (
+            self.server_args.elastic_ep_backend is not None
+            and self.server_args.ep_join_mode == "recover"
+        ):
+            model_runner.post_capture_elastic_ep_recover()
+
         # Dispatch the model worker
         if self.spec_algorithm.is_none():
             self.model_worker = self.tp_worker

python/sglang/srt/managers/tp_worker.py

@@ -342,8 +342,8 @@ class TpModelWorker(BaseTpWorker):
         self.world_group = get_world_group()
 
         # Sync random seed across TP workers.
-        # Scale joiners cannot enter the launch-time WORLD broadcast.
-        if server_args.is_ep_scale_joiner:
+        # Elastic joiners cannot enter the launch-time WORLD broadcast.
+        if server_args.is_ep_joiner:
             self.random_seed = server_args.random_seed
         else:
             self.random_seed = broadcast_pyobj(