### 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.
ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Changing scene can only be done from the main thread.");
	if (current_scene) {
		memdelete(current_scene);
		current_scene = nullptr;
	}
	
	if (unlikely(_quit)) {
		if (p_to) { 
			memdelete(p_to);
		}
		return;
	}
	if (p_to) {
		current_scene = p_to;
		root->add_child(p_to);
		
		root->update_mouse_cursor_state();
	}
}
Error SceneTree::change_scene_to_file(const String &p_path) {
	ERR_FAIL_COND_V_MSG(!Thread::is_main_thread(), ERR_INVALID_PARAMETER, "Changing scene can only be done from the main thread.");
	Ref<PackedScene> new_scene = ResourceLoader::load(p_path);
	if (new_scene.is_null()) {
		return ERR_CANT_OPEN;
	}
	return change_scene_to_packed(new_scene);
}
Error SceneTree::change_scene_to_packed(const Ref<PackedScene> &p_scene) {