### 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)

return len_fade_from;
}
float AnimationNodeStateMachinePlayback::get_fading_time() const {
	return fading_time;
}
float AnimationNodeStateMachinePlayback::get_fading_pos() const {
	return fading_pos;
}
void AnimationNodeStateMachinePlayback::_clear_path_children(AnimationTree *p_tree, AnimationNodeStateMachine *p_state_machine, bool p_test_only) {
	List<AnimationNode::ChildNode> child_nodes;
	p_state_machine->get_child_nodes(&child_nodes);
	for (int i = 0; i < child_nodes.size(); i++) {
		Ref<AnimationNodeStateMachine> anodesm = child_nodes[i].node;
		if (anodesm.is_valid() && anodesm->get_state_machine_type() == AnimationNodeStateMachine::STATE_MACHINE_TYPE_GROUPED) {
			Ref<AnimationNodeStateMachinePlayback> playback = p_tree->get(base_path + child_nodes[i].name + "/playback");
			ERR_FAIL_COND(!playback.is_valid());
			playback->_set_base_path(base_path + child_nodes[i].name + "/");
			if (p_test_only) {
				playback = playback->duplicate();
			}
			playback->path.clear();
			playback->_clear_path_children(p_tree, anodesm.ptr(), p_test_only);
			if (current != child_nodes[i].name) {
				playback->_start(anodesm.ptr()); 
			}
		}
	}
}
void AnimationNodeStateMachinePlayback::_start_children(AnimationTree *p_tree, AnimationNodeStateMachine *p_state_machine, const String &p_path, bool p_test_only) {