### Godot version

4.2

### System information

Linux

### Issue description

A nested state machine isn't traveling to its animation in `_ready()` in 4.2 as it was in 4.1.3.

In the ready function, we make the `root` state machine travel to the state of the `Stages` nested state machine. Then we immediately call the `Stages` nested state machine to travel to `Stage2`. This works fine in Godot v4.1.3, but not in Godot v4.2, where it seems a delay is first needed for the second travel animation to work.

![image](https://github.com/godotengine/godot/assets/788630/f1582a99-a68e-45d9-82b8-8367e62a9cba)
![image](https://github.com/godotengine/godot/assets/788630/c7c1f9fa-c78b-4259-bd36-81ebd8cef9aa)

### Steps to reproduce

Run the MRP below in 4.1.3 and 4.2.

In 4.1.3, the animation tree will reach `Stage2` and show Finished on the screen.

In 4.2, the animation tree will only reach `Stage1` and show Stuck on the screen.

### Minimal reproduction project

[MRP-AnimationNodeStateMachinePlayback.zip](https://github.com/godotengine/godot/files/13533132/MRP-AnimationNodeStateMachinePlayback.zip)

playback->_start(anodesm.ptr());
			}
			ChildStateMachineInfo child_info;
			child_info.playback = playback;
			
			bool child_found_route = true;
			bool is_current_same_state = temp_path[i] == playback->get_current_node();
			if (!is_parent_same_state) {
				
				String child_path = "/" + playback->get_current_node();
				while (true) {
					Ref<AnimationNodeStateMachine> child_anodesm = p_state_machine->find_node_by_path(concatenated + child_path);
					if (!child_anodesm.is_valid() || child_anodesm->get_state_machine_type() != AnimationNodeStateMachine::STATE_MACHINE_TYPE_GROUPED) {
						break;
					}
					Ref<AnimationNodeStateMachinePlayback> child_playback = p_tree->get(base_path + concatenated + child_path + "/playback");
					ERR_FAIL_COND_V(!child_playback.is_valid(), false);
					child_playback->_set_base_path(base_path + concatenated + "/");
					if (p_test_only) {
						child_playback = child_playback->duplicate();
					}
					child_playback->_travel_main("End");
					child_found_route &= child_playback->_travel(p_tree, child_anodesm.ptr(), false, p_test_only);
					child_path += "/" + child_playback->get_current_node();
				}
				
				playback->_start(anodesm.ptr());
			}
			is_parent_same_state = is_current_same_state;
			bool is_deepest_state = i == temp_path.size() - 1;