### 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_
	SelfList<Node> *n = xform_change_list.first();
	while (n) {
		Node *node = n->self();
		SelfList<Node> *nx = n->next();
		xform_change_list.remove(n);
		n = nx;
		node->notification(NOTIFICATION_TRANSFORM_CHANGED);
	}
}
void SceneTree::_flush_ugc() {
	ugc_locked = true;
	while (unique_group_calls.size()) {
		HashMap<UGCall, Vector<Variant>, UGCall>::Iterator E = unique_group_calls.begin();
		const Variant **argptrs = (const Variant **)alloca(E->value.size() * sizeof(Variant *));
		for (int i = 0; i < E->value.size(); i++) {
			argptrs[i] = &E->value[i];
		}
		call_group_flagsp(GROUP_CALL_DEFAULT, E->key.group, E->key.call, argptrs, E->value.size());
		unique_group_calls.remove(E);
	}
	ugc_locked = false;
}
void SceneTree::_update_group_order(Group &g) {