### 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)

process_state->is_testing = p_test_only;
	return _process(p_playback_info, p_test_only);
}
double AnimationNode::_process(const AnimationMixer::PlaybackInfo p_playback_info, bool p_test_only) {
	double ret = 0;
	GDVIRTUAL_CALL(_process, p_playback_info.time, p_playback_info.seeked, p_playback_info.is_external_seeking, p_test_only, ret);
	return ret;
}
void AnimationNode::set_filter_path(const NodePath &p_path, bool p_enable) {
	if (p_enable) {
		filter[p_path] = true;
	} else {
		filter.erase(p_path);
	}
}
void AnimationNode::set_filter_enabled(bool p_enable) {
	filter_enabled = p_enable;
}
bool AnimationNode::is_filter_enabled() const {
	return filter_enabled;
}
void AnimationNode::set_closable(bool p_closable) {
	closable = p_closable;
}
bool AnimationNode::is_closable() const {