### 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.
accept_quit = p_enable;
}
bool SceneTree::is_quit_on_go_back() const {
	return quit_on_go_back;
}
void SceneTree::set_quit_on_go_back(bool p_enable) {
	quit_on_go_back = p_enable;
}
#ifdef TOOLS_ENABLED
bool SceneTree::is_node_being_edited(const Node *p_node) const {
	return Engine::get_singleton()->is_editor_hint() && edited_scene_root && (edited_scene_root->is_ancestor_of(p_node) || edited_scene_root == p_node);
}
#endif
#ifdef DEBUG_ENABLED
void SceneTree::set_debug_collisions_hint(bool p_enabled) {
	debug_collisions_hint = p_enabled;
}
bool SceneTree::is_debugging_collisions_hint() const {
	return debug_collisions_hint;
}
void SceneTree::set_debug_paths_hint(bool p_enabled) {
	debug_paths_hint = p_enabled;
}
bool SceneTree::is_debugging_paths_hint() const {
	return debug_paths_hint;
}
void SceneTree::set_debug_navigation_hint(bool p_enabled) {
	debug_navigation_hint = p_enabled;
}
bool SceneTree::is_debugging_navigation_hint() const {