### 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.
return paused;
}
void SceneTree::_process_group(ProcessGroup *p_group, bool p_physics) {
	
	
	p_group->call_queue.flush(); 
	Vector<Node *> &nodes = p_physics ? p_group->physics_nodes : p_group->nodes;
	if (nodes.is_empty()) {
		return;
	}
	if (p_physics) {
		if (p_group->physics_node_order_dirty) {
			nodes.sort_custom<Node::ComparatorWithPhysicsPriority>();
			p_group->physics_node_order_dirty = false;
		}
	} else {
		if (p_group->node_order_dirty) {
			nodes.sort_custom<Node::ComparatorWithPriority>();
			p_group->node_order_dirty = false;
		}
	}
	
	Vector<Node *> nodes_copy = nodes;
	uint32_t node_count = nodes_copy.size();