### 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)

clear_caches();
		return;
	}
	AnimationMixer *mixer = Object::cast_to<AnimationMixer>(get_node_or_null(animation_player));
	if (mixer) {
		if (!mixer->is_connected(SNAME("caches_cleared"), callable_mp(this, &AnimationTree::_setup_animation_player))) {
			mixer->connect(SNAME("caches_cleared"), callable_mp(this, &AnimationTree::_setup_animation_player), CONNECT_DEFERRED);
		}
		if (!mixer->is_connected(SNAME("animation_list_changed"), callable_mp(this, &AnimationTree::_setup_animation_player))) {
			mixer->connect(SNAME("animation_list_changed"), callable_mp(this, &AnimationTree::_setup_animation_player), CONNECT_DEFERRED);
		}
		Node *root = mixer->get_node_or_null(mixer->get_root_node());
		if (root) {
			set_root_node(get_path_to(root, true));
		}
		while (animation_libraries.size()) {
			remove_animation_library(animation_libraries[0].name);
		}
		List<StringName> list;
		mixer->get_animation_library_list(&list);
		for (int i = 0; i < list.size(); i++) {
			Ref<AnimationLibrary> lib = mixer->get_animation_library(list[i]);
			if (lib.is_valid()) {
				add_animation_library(list[i], lib);
			}
		}
	}
	clear_caches();
}
void AnimationTree::_validate_property(PropertyInfo &p_property) const {