### 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.
node_threading_disabled = p_disable;
}
SceneTree::SceneTree() {
	if (singleton == nullptr) {
		singleton = this;
	}
	debug_collisions_color = GLOBAL_DEF("debug/shapes/collision/shape_color", Color(0.0, 0.6, 0.7, 0.42));
	debug_collision_contact_color = GLOBAL_DEF("debug/shapes/collision/contact_color", Color(1.0, 0.2, 0.1, 0.8));
	debug_paths_color = GLOBAL_DEF("debug/shapes/paths/geometry_color", Color(0.1, 1.0, 0.7, 0.4));
	debug_paths_width = GLOBAL_DEF("debug/shapes/paths/geometry_width", 2.0);
	collision_debug_contacts = GLOBAL_DEF(PropertyInfo(Variant::INT, "debug/shapes/collision/max_contacts_displayed", PROPERTY_HINT_RANGE, "0,20000,1"), 10000);
	GLOBAL_DEF("debug/shapes/collision/draw_2d_outlines", true);
	process_group_call_queue_allocator = memnew(CallQueue::Allocator(64));
	Math::randomize();
	
	root = memnew(Window);
	root->set_min_size(Size2i(64, 64)); 
	root->set_process_mode(Node::PROCESS_MODE_PAUSABLE);
	root->set_name("root");
	root->set_title(GLOBAL_GET("application/config/name"));
	if (Engine::get_singleton()->is_editor_hint()) {
		root->set_wrap_controls(true);
	}
#ifndef _3D_DISABLED
	if (!root->get_world_3d().is_valid()) {
		root->set_world_3d(Ref<World3D>(memnew(World3D)));