### 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)

open_list.erase(least_cost_transition);
	}
	
	if (found_route) {
		
		StringName at = travel;
		while (at != current) {
			new_path.push_back(at);
			at = cost_map[at].prev;
		}
		new_path.reverse();
		
		
		
		String current_path = current;
		int len = new_path.size() + 1;
		for (int i = 0; i < len; i++) {
			Ref<AnimationNodeStateMachine> anodesm = p_state_machine->find_node_by_path(current_path);
			if (anodesm.is_valid() && anodesm->get_state_machine_type() == AnimationNodeStateMachine::STATE_MACHINE_TYPE_GROUPED) {
				Ref<AnimationNodeStateMachinePlayback> playback = p_tree->get(base_path + current_path + "/playback");
				ERR_FAIL_COND_V(!playback.is_valid(), false);
				playback->_set_base_path(base_path + current_path + "/");
				if (p_test_only) {
					playback = playback->duplicate();
				}
				if (i > 0) {
					playback->_start(anodesm.ptr());
				}
				if (i >= new_path.size()) {
					break;