Remove unused and deprecated api (#107292)

Leftover cleanup work.

Related to #105676 and #105746

Author: volodk85

server/src/main/java/org/elasticsearch/persistent/PersistentTasksService.java

@@ -49,19 +49,6 @@ public PersistentTasksService(ClusterService clusterService, ThreadPool threadPo
         this.threadPool = threadPool;
     }
 
-    /**
-     * Notifies the master node to create new persistent task and to assign it to a node.
-     */
-    @Deprecated(forRemoval = true)
-    public <Params extends PersistentTaskParams> void sendStartRequest(
-        final String taskId,
-        final String taskName,
-        final Params taskParams,
-        final ActionListener<PersistentTask<Params>> listener
-    ) {
-        sendStartRequest(taskId, taskName, taskParams, null, listener);
-    }
-
     /**
      * Notifies the master node to create new persistent task and to assign it to a node. Accepts operation timeout as optional parameter
      */
@@ -156,11 +143,6 @@ void sendUpdateStateRequest(
         execute(request, UpdatePersistentTaskStatusAction.INSTANCE, listener);
     }
 
-    @Deprecated(forRemoval = true)
-    public void sendRemoveRequest(final String taskId, final ActionListener<PersistentTask<?>> listener) {
-        sendRemoveRequest(taskId, null, listener);
-    }
-
     /**
      * Notifies the master node to remove a persistent task from the cluster state. Accepts operation timeout as optional parameter
      */