### 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)

if (!properties_dirty) {
		return;
	}
	properties.clear();
	property_reference_map.clear();
	property_parent_map.clear();
	input_activity_map.clear();
	input_activity_map_get.clear();
	if (root_animation_node.is_valid()) {
		_update_properties_for_node(SceneStringNames::get_singleton()->parameters_base_path, root_animation_node);
	}
	properties_dirty = false;
	notify_property_list_changed();
}
void AnimationTree::_notification(int p_what) {
	switch (p_what) {
		case NOTIFICATION_ENTER_TREE: {
			_setup_animation_player();
			if (active) {
				_set_process(true);
			}
		} break;
	}
}
void AnimationTree::set_animation_player(const NodePath &p_path) {