### 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.
#include "core/io/image_loader.h"
#include "core/io/marshalls.h"
#include "core/io/resource_loader.h"
#include "core/object/message_queue.h"
#include "core/object/worker_thread_pool.h"
#include "core/os/keyboard.h"
#include "core/os/os.h"
#include "core/string/print_string.h"
#include "node.h"
#include "scene/animation/tween.h"
#include "scene/debugger/scene_debugger.h"
#include "scene/gui/control.h"
#include "scene/main/multiplayer_api.h"
#include "scene/main/viewport.h"
#include "scene/resources/environment.h"
#include "scene/resources/font.h"
#include "scene/resources/material.h"
#include "scene/resources/mesh.h"
#include "scene/resources/packed_scene.h"
#include "scene/resources/world_2d.h"
#include "scene/resources/world_3d.h"
#include "scene/scene_string_names.h"
#include "servers/display_server.h"
#include "servers/navigation_server_3d.h"
#include "servers/physics_server_2d.h"
#include "servers/physics_server_3d.h"
#include "window.h"
#include <stdio.h>
#include <stdlib.h>
void SceneTreeTimer::_bind_methods() {