### 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)

ERR_FAIL_COND(!states.has(p_name));
	ERR_FAIL_COND(states.has(p_new_name));
	ERR_FAIL_COND(!can_edit_node(p_name));
	states[p_new_name] = states[p_name];
	states.erase(p_name);
	_rename_transitions(p_name, p_new_name);
	emit_signal(SNAME("animation_node_renamed"), get_instance_id(), p_name, p_new_name);
	emit_changed();
	emit_signal(SNAME("tree_changed"));
}
void AnimationNodeStateMachine::_rename_transitions(const StringName &p_name, const StringName &p_new_name) {
	if (updating_transitions) {
		return;
	}
	updating_transitions = true;
	for (int i = 0; i < transitions.size(); i++) {
		if (transitions[i].from == p_name) {
			transitions.write[i].from = p_new_name;
		}
		if (transitions[i].to == p_name) {
			transitions.write[i].to = p_new_name;
		}
	}
	updating_transitions = false;
}
void AnimationNodeStateMachine::get_node_list(List<StringName> *r_nodes) const {