### 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.
Node::current_process_thread_group = local_process_group_cache[p_index]->owner;
	_process_group(local_process_group_cache[p_index], p_physics);
	Node::current_process_thread_group = nullptr;
}
void SceneTree::_process(bool p_physics) {
	if (process_groups_dirty) {
		{
			
			
			ProcessGroup **pg_ptr = (ProcessGroup **)process_groups.ptr(); 
			uint32_t pg_count = process_groups.size();
			for (uint32_t i = 0; i < pg_count; i++) {
				if (pg_ptr[i]->removed) {
					
					pg_ptr[i] = pg_ptr[pg_count - 1];