### 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_
	HashMap<StringName, Group>::Iterator E = group_map.find(p_group);
	if (!E) {
		E = group_map.insert(p_group, Group());
	}
	ERR_FAIL_COND_V_MSG(E->value.nodes.has(p_node), &E->value, "Already in group: " + p_group + ".");
	E->value.nodes.push_back(p_node);
	
	E->value.changed = true;
	return &E->value;
}
void SceneTree::remove_from_group(const StringName &p_group, Node *p_node) {
	_THREAD_SAFE_METHOD_
	HashMap<StringName, Group>::Iterator E = group_map.find(p_group);
	ERR_FAIL_COND(!E);
	E->value.nodes.erase(p_node);
	if (E->value.nodes.is_empty()) {
		group_map.remove(E);
	}
}
void SceneTree::make_group_changed(const StringName &p_group) {
	_THREAD_SAFE_METHOD_
	HashMap<StringName, Group>::Iterator E = group_map.find(p_group);
	if (E) {
		E->value.changed = true;
	}
}
void SceneTree::flush_transform_notifications() {