### 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)

PackedStringArray warnings = Node::get_configuration_warnings();
	if (!root_animation_node.is_valid()) {
		warnings.push_back(RTR("No root AnimationNode for the graph is set."));
	}
	return warnings;
}
void AnimationTree::_tree_changed() {
	if (properties_dirty) {
		return;
	}
	call_deferred(SNAME("_update_properties"));
	properties_dirty = true;
}
void AnimationTree::_animation_node_renamed(const ObjectID &p_oid, const String &p_old_name, const String &p_new_name) {
	ERR_FAIL_COND(!property_reference_map.has(p_oid));
	String base_path = property_reference_map[p_oid];
	String old_base = base_path + p_old_name;
	String new_base = base_path + p_new_name;
	for (const PropertyInfo &E : properties) {
		if (E.name.begins_with(old_base)) {
			String new_name = E.name.replace_first(old_base, new_base);
			property_map[new_name] = property_map[E.name];
			property_map.erase(E.name);
		}
	}
	// 
	properties_dirty = true;
	_update_properties();
}
void AnimationTree::_animation_node_removed(const ObjectID &p_oid, const StringName &p_node) {