### Godot version

4.2-rc1

### System information

Windows 11

### Issue description

The issue does not exist on Godot 4.1.3, only in 4.2

When you call `seek()` on the AnimationPlayer, the animation does not update instantly and freezes for a brief moment.
So if you need to switch multiple animations in a short amount of time while also calling `seek()`, the animation will just stop.

### Steps to reproduce

- Add a Sprite2D
- Set a spritesheet for the Sprite2D
- Add an AnimationPlayer
- Create a Property animation track and add some frames
- Loop the animation
- Create a different animation for the same sprite an loop it
- Switch between animations and call `seek()` when you do

### Minimal reproduction project

The included project demonstrates the issue perfectly (I know that calling `play()` and `seek()` on each `_process()` is a bit overkill, but it does allow for a better demonstration). The same project works perfectly in Godot 4.1.3.

[AnimationBug.zip](https://github.com/godotengine/godot/files/13400011/AnimationBug.zip)

String name = p_name;
	if (name.begins_with("playback/play")) { 
		set_current_animation(p_value);
	} else if (name.begins_with("next/")) {
		String which = name.get_slicec('/', 1);
		animation_set_next(which, p_value);
	} else if (p_name == SceneStringNames::get_singleton()->blend_times) {
		Array array = p_value;
		int len = array.size();
		ERR_FAIL_COND_V(len % 3, false);
		for (int i = 0; i < len / 3; i++) {
			StringName from = array[i * 3 + 0];
			StringName to = array[i * 3 + 1];
			float time = array[i * 3 + 2];
			set_blend_time(from, to, time);
		}
#ifndef DISABLE_DEPRECATED
	} else if (p_name == "method_call_mode") {
		set_callback_mode_method(static_cast<AnimationCallbackModeMethod>((int)p_value));
	} else if (p_name == "playback_process_mode") {
		set_callback_mode_process(static_cast<AnimationCallbackModeProcess>((int)p_value));
	} else if (p_name == "playback_active") {
		set_active(p_value);
#endif 
	} else {
		return false;
	}
	return true;
}
bool AnimationPlayer::_get(const StringName &p_name, Variant &r_ret) const {