### 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.
_flush_delete_queue();
	_flush_ugc();
	initialized = false;
	MainLoop::finalize();
	if (root) {
		root->_set_tree(nullptr);
		root->_propagate_after_exit_tree();
		memdelete(root); 
		root = nullptr;
	}
	
	
	_flush_delete_queue();
	
	for (Ref<SceneTreeTimer> &timer : timers) {
		timer->release_connections();
	}
	timers.clear();
	
	for (Ref<Tween> &tween : tweens) {
		tween->clear();
	}
	tweens.clear();
}
void SceneTree::quit(int p_exit_code) {