### 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_INDEX(p_index, inputs.size());
	inputs.remove_at(p_index);
	emit_changed();
}
bool AnimationNode::set_input_name(int p_input, const String &p_name) {
	ERR_FAIL_INDEX_V(p_input, inputs.size(), false);
	ERR_FAIL_COND_V(p_name.contains(".") || p_name.contains("/"), false);
	inputs.write[p_input].name = p_name;
	emit_changed();
	return true;
}
String AnimationNode::get_input_name(int p_input) const {
	ERR_FAIL_INDEX_V(p_input, inputs.size(), String());
	return inputs[p_input].name;
}
int AnimationNode::get_input_count() const {
	return inputs.size();
}
int AnimationNode::find_input(const String &p_name) const {
	int idx = -1;
	for (int i = 0; i < inputs.size(); i++) {
		if (inputs[i].name == p_name) {
			idx = i;
			break;
		}
	}
	return idx;
}
double AnimationNode::process(const AnimationMixer::PlaybackInfo p_playback_info, bool p_test_only) {