### 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_NULL(process_state);
	p_playback_info.track_weights = node_state.track_weights;
	process_state->tree->make_animation_instance(p_animation, p_playback_info);
}
double AnimationNode::_pre_process(ProcessState *p_process_state, AnimationMixer::PlaybackInfo p_playback_info) {
	process_state = p_process_state;
	double t = process(p_playback_info);
	process_state = nullptr;
	return t;
}
void AnimationNode::make_invalid(const String &p_reason) {
	ERR_FAIL_NULL(process_state);
	process_state->valid = false;
	if (!process_state->invalid_reasons.is_empty()) {
		process_state->invalid_reasons += "\n";
	}
	process_state->invalid_reasons += String::utf8("•  ") + p_reason;
}
AnimationTree *AnimationNode::get_animation_tree() const {