### 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)

_update_properties(); // 
	if (!root_animation_node.is_valid()) {
		return false;
	}
	{ // 
		process_pass++;
		// 
		process_state = AnimationNode::ProcessState();
		process_state.tree = this;
		process_state.valid = true;
		process_state.invalid_reasons = "";
		process_state.last_pass = process_pass;
		process_state.track_map = p_track_map;
		// 
		root_animation_node->node_state.track_weights.resize(p_track_count);
		real_t *src_blendsw = root_animation_node->node_state.track_weights.ptrw();
		for (int i = 0; i < p_track_count; i++) {
			src_blendsw[i] = 1.0; // 
		}
		root_animation_node->node_state.base_path = SceneStringNames::get_singleton()->parameters_base_path;
		root_animation_node->node_state.parent = nullptr;
	}
	// 
	{
		PlaybackInfo pi;
		if (started) {
			// 
			pi.seeked = true;
			root_animation_node->_pre_process(&process_state, pi);
			started = false;