### 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)

for (int i = 0; i < transitions.size(); i++) {
		if (transitions[i].transition == p_transition) {
			remove_transition_by_index(i);
			return;
		}
	}
}
void AnimationNodeStateMachine::set_graph_offset(const Vector2 &p_offset) {
	graph_offset = p_offset;
}
Vector2 AnimationNodeStateMachine::get_graph_offset() const {
	return graph_offset;
}
double AnimationNodeStateMachine::_process(const AnimationMixer::PlaybackInfo p_playback_info, bool p_test_only) {
	Ref<AnimationNodeStateMachinePlayback> playback_new = get_parameter(playback);
	ERR_FAIL_COND_V(playback_new.is_null(), 0.0);
	playback_new->_set_grouped(state_machine_type == STATE_MACHINE_TYPE_GROUPED);
	if (p_test_only) {
		playback_new = playback_new->duplicate(); 
	}
	return playback_new->process(node_state.base_path, this, p_playback_info, p_test_only);
}
String AnimationNodeStateMachine::get_caption() const {
	return "StateMachine";
}
Ref<AnimationNode> AnimationNodeStateMachine::get_child_by_name(const StringName &p_name) const {
	return get_node(p_name);
}
bool AnimationNodeStateMachine::_set(const StringName &p_name, const Variant &p_value) {