### 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_COND_EDMSG(is_grouped, "Grouped AnimationNodeStateMachinePlayback must be handled by parent AnimationNodeStateMachinePlayback. You need to retrieve the parent Root/Nested AnimationNodeStateMachine.");
	ERR_FAIL_COND_EDMSG(String(p_state).contains("/Start") || String(p_state).contains("/End"), "Grouped AnimationNodeStateMachinePlayback doesn't allow to play Start/End directly. Instead, play the prev or next state of group in the parent AnimationNodeStateMachine.");
	_start_main(p_state, p_reset);
}
void AnimationNodeStateMachinePlayback::next() {
	ERR_FAIL_COND_EDMSG(is_grouped, "Grouped AnimationNodeStateMachinePlayback must be handled by parent AnimationNodeStateMachinePlayback. You need to retrieve the parent Root/Nested AnimationNodeStateMachine.");
	_next_main();
}
void AnimationNodeStateMachinePlayback::stop() {
	ERR_FAIL_COND_EDMSG(is_grouped, "Grouped AnimationNodeStateMachinePlayback must be handled by parent AnimationNodeStateMachinePlayback. You need to retrieve the parent Root/Nested AnimationNodeStateMachine.");
	_stop_main();
}
void AnimationNodeStateMachinePlayback::_travel_main(const StringName &p_state, bool p_reset_on_teleport) {
	travel_request = p_state;
	reset_request_on_teleport = p_reset_on_teleport;
	stop_request = false;
}
void AnimationNodeStateMachinePlayback::_start_main(const StringName &p_state, bool p_reset) {
	travel_request = StringName();
	path.clear();
	reset_request = p_reset;
	start_request = p_state;
	stop_request = false;
}
void AnimationNodeStateMachinePlayback::_next_main() {
	next_request = true;
}
void AnimationNodeStateMachinePlayback::_stop_main() {