### 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_V_MSG(p_scene.is_null(), ERR_INVALID_PARAMETER, "Can't change to a null scene. Use unload_current_scene() if you wish to unload it.");
	Node *new_scene = p_scene->instantiate();
	ERR_FAIL_NULL_V(new_scene, ERR_CANT_CREATE);
	call_deferred(SNAME("_change_scene"), new_scene);
	return OK;
}
Error SceneTree::reload_current_scene() {
	ERR_FAIL_COND_V_MSG(!Thread::is_main_thread(), ERR_INVALID_PARAMETER, "Reloading scene can only be done from the main thread.");
	ERR_FAIL_NULL_V(current_scene, ERR_UNCONFIGURED);
	String fname = current_scene->get_scene_file_path();
	return change_scene_to_file(fname);
}
void SceneTree::unload_current_scene() {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Unloading the current scene can only be done from the main thread.");
	if (current_scene) {
		memdelete(current_scene);
		current_scene = nullptr;
	}
}
void SceneTree::add_current_scene(Node *p_current) {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Adding a current scene can only be done from the main thread.");
	current_scene = p_current;
	root->add_child(p_current);
}
Ref<SceneTreeTimer> SceneTree::create_timer(double p_delay_sec, bool p_process_always, bool p_process_in_physics, bool p_ignore_time_scale) {