### 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->_travel_main("End");
				if (!playback->_travel(p_tree, anodesm.ptr(), false, p_test_only)) {
					found_route = false;
					break;
				}
			}
			if (i >= new_path.size()) {
				break; 
			}
			current_path = new_path[i];
		}
	}
	
	if (found_route) {
		r_path = new_path;
		return true;
	} else {
		return false;
	}
}
double AnimationNodeStateMachinePlayback::process(const String &p_base_path, AnimationNodeStateMachine *p_state_machine, const AnimationMixer::PlaybackInfo p_playback_info, bool p_test_only) {
	double rem = _process(p_base_path, p_state_machine, p_playback_info, p_test_only);
	start_request = StringName();
	next_request = false;
	stop_request = false;
	reset_request_on_teleport = false;
	return rem;
}
double AnimationNodeStateMachinePlayback::_process(const String &p_base_path, AnimationNodeStateMachine *p_state_machine, const AnimationMixer::PlaybackInfo p_playback_info, bool p_test_only) {