### 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.
Vector<int> indices;
	for (int i = 0; i < 8 * 3; i++) {
		indices.push_back(diamond_faces[i]);
	}
	Vector<Vector3> vertices;
	for (int i = 0; i < 6; i++) {
		vertices.push_back(diamond[i] * 0.1);
	}
	Array arr;
	arr.resize(Mesh::ARRAY_MAX);
	arr[Mesh::ARRAY_VERTEX] = vertices;
	arr[Mesh::ARRAY_INDEX] = indices;
	debug_contact_mesh->add_surface_from_arrays(Mesh::PRIMITIVE_TRIANGLES, arr);
	debug_contact_mesh->surface_set_material(0, mat);
	return debug_contact_mesh;
}
void SceneTree::set_pause(bool p_enabled) {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Pause can only be set from the main thread.");
	if (p_enabled == paused) {
		return;
	}
	paused = p_enabled;
	PhysicsServer3D::get_singleton()->set_active(!p_enabled);
	PhysicsServer2D::get_singleton()->set_active(!p_enabled);
	if (get_root()) {
		get_root()->_propagate_pause_notification(p_enabled);
	}
}
bool SceneTree::is_paused() const {