### 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_
	if (debug_paths_material.is_valid()) {
		return debug_paths_material;
	}
	Ref<StandardMaterial3D> _debug_material = Ref<StandardMaterial3D>(memnew(StandardMaterial3D));
	_debug_material->set_shading_mode(StandardMaterial3D::SHADING_MODE_UNSHADED);
	_debug_material->set_transparency(StandardMaterial3D::TRANSPARENCY_ALPHA);
	_debug_material->set_flag(StandardMaterial3D::FLAG_SRGB_VERTEX_COLOR, true);
	_debug_material->set_flag(StandardMaterial3D::FLAG_ALBEDO_FROM_VERTEX_COLOR, true);
	_debug_material->set_albedo(get_debug_paths_color());
	debug_paths_material = _debug_material;
	return debug_paths_material;
}
Ref<Material> SceneTree::get_debug_collision_material() {
	_THREAD_SAFE_METHOD_
	if (collision_material.is_valid()) {
		return collision_material;
	}
	Ref<StandardMaterial3D> line_material = Ref<StandardMaterial3D>(memnew(StandardMaterial3D));
	line_material->set_shading_mode(StandardMaterial3D::SHADING_MODE_UNSHADED);
	line_material->set_transparency(StandardMaterial3D::TRANSPARENCY_ALPHA);
	line_material->set_flag(StandardMaterial3D::FLAG_SRGB_VERTEX_COLOR, true);
	line_material->set_flag(StandardMaterial3D::FLAG_ALBEDO_FROM_VERTEX_COLOR, true);
	line_material->set_albedo(get_debug_collisions_color());
	collision_material = line_material;
	return collision_material;
}
Ref<ArrayMesh> SceneTree::get_debug_contact_mesh() {