### Godot version

4.1 dev3

### System information

Windows 11, Forward+, Intel 13700k, 4070ti

### Issue description

In 4.0.x and in 4.1 dev2, if you overwrite the rotation of a bone in `_process` it has preferred over the currently running animation. In 4.1 dev3, the running animation takes precedence and overwrites any changes in `_process`. This happens both with a simple animation and with an animation tree, and happens in-editor and in-game.

### Steps to reproduce

1. Open the attached project.
2. Click on `AnimationPlayer` and pick an animation, e.g. `Armature|Standing`. Set it to looping if it's not already. (@RandomShaper's note: and play it).
3. Click on the `PointOfInterest` and move it along the X axis.
4. If you're on:
    - 4.0.x or 4.1-dev2: Notice the head rotating
    - 4.1-dev3: Notice that nothing happens

### Minimal reproduction project

[BoneRotationIssue-4.0.3.zip](https://github.com/godotengine/godot/files/11582187/BoneRotationIssue-4.0.3.zip)
[BoneRotationIssue-4.1-dev.zip](https://github.com/godotengine/godot/files/11582188/BoneRotationIssue-4.1-dev.zip)


Should include all licensing stuff for the model, but let me just add [the link](https://opengameart.org/content/human-model-ready-for-animation) for good measure.
_THREAD_SAFE_METHOD_
	
	List<Ref<Tween>>::Element *L = tweens.back();
	for (List<Ref<Tween>>::Element *E = tweens.front(); E;) {
		List<Ref<Tween>>::Element *N = E->next();
		
		if (!E->get()->can_process(paused) || (p_physics == (E->get()->get_process_mode() == Tween::TWEEN_PROCESS_IDLE))) {
			if (E == L) {
				break;
			}
			E = N;
			continue;
		}
		if (!E->get()->step(p_delta)) {
			E->get()->clear();
			tweens.erase(E);
		}
		if (E == L) {
			break;
		}
		E = N;
	}
}
void SceneTree::finalize() {