### 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)

AnimationMixer::_rename_animation(p_from_name, p_to_name);
	
	List<BlendKey> to_erase;
	HashMap<BlendKey, double, BlendKey> to_insert;
	for (const KeyValue<BlendKey, double> &E : blend_times) {
		BlendKey bk = E.key;
		BlendKey new_bk = bk;
		bool erase = false;
		if (bk.from == p_from_name) {
			new_bk.from = p_to_name;
			erase = true;
		}
		if (bk.to == p_from_name) {
			new_bk.to = p_to_name;
			erase = true;
		}
		if (erase) {
			to_erase.push_back(bk);
			to_insert[new_bk] = E.value;
		}
	}