### 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_
	ERR_FAIL_NULL(p_object);
	p_object->_is_queued_for_deletion = true;
	delete_queue.push_back(p_object->get_instance_id());
}
int SceneTree::get_node_count() const {
	return nodes_in_tree_count;
}
void SceneTree::set_edited_scene_root(Node *p_node) {
#ifdef TOOLS_ENABLED
	edited_scene_root = p_node;
#endif
}
Node *SceneTree::get_edited_scene_root() const {
#ifdef TOOLS_ENABLED
	return edited_scene_root;
#else
	return nullptr;
#endif
}
void SceneTree::set_current_scene(Node *p_scene) {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Changing scene can only be done from the main thread.");
	ERR_FAIL_COND(p_scene && p_scene->get_parent() != root);
	current_scene = p_scene;
}
Node *SceneTree::get_current_scene() const {
	return current_scene;
}
void SceneTree::_change_scene(Node *p_to) {