### 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.
MethodInfo mi;
	mi.name = "call_group_flags";
	mi.arguments.push_back(PropertyInfo(Variant::INT, "flags"));
	mi.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "group"));
	mi.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "method"));
	ClassDB::bind_vararg_method(METHOD_FLAGS_DEFAULT, "call_group_flags", &SceneTree::_call_group_flags, mi);
	ClassDB::bind_method(D_METHOD("notify_group_flags", "call_flags", "group", "notification"), &SceneTree::notify_group_flags);
	ClassDB::bind_method(D_METHOD("set_group_flags", "call_flags", "group", "property", "value"), &SceneTree::set_group_flags);
	MethodInfo mi2;
	mi2.name = "call_group";
	mi2.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "group"));
	mi2.arguments.push_back(PropertyInfo(Variant::STRING_NAME, "method"));
	ClassDB::bind_vararg_method(METHOD_FLAGS_DEFAULT, "call_group", &SceneTree::_call_group, mi2);
	ClassDB::bind_method(D_METHOD("notify_group", "group", "notification"), &SceneTree::notify_group);
	ClassDB::bind_method(D_METHOD("set_group", "group", "property", "value"), &SceneTree::set_group);
	ClassDB::bind_method(D_METHOD("get_nodes_in_group", "group"), &SceneTree::_get_nodes_in_group);
	ClassDB::bind_method(D_METHOD("get_first_node_in_group", "group"), &SceneTree::get_first_node_in_group);
	ClassDB::bind_method(D_METHOD("set_current_scene", "child_node"), &SceneTree::set_current_scene);
	ClassDB::bind_method(D_METHOD("get_current_scene"), &SceneTree::get_current_scene);
	ClassDB::bind_method(D_METHOD("change_scene_to_file", "path"), &SceneTree::change_scene_to_file);
	ClassDB::bind_method(D_METHOD("change_scene_to_packed", "packed_scene"), &SceneTree::change_scene_to_packed);
	ClassDB::bind_method(D_METHOD("reload_current_scene"), &SceneTree::reload_current_scene);
	ClassDB::bind_method(D_METHOD("unload_current_scene"), &SceneTree::unload_current_scene);
	ClassDB::bind_method(D_METHOD("_change_scene"), &SceneTree::_change_scene);
	ClassDB::bind_method(D_METHOD("set_multiplayer", "multiplayer", "root_path"), &SceneTree::set_multiplayer, DEFVAL(NodePath()));
	ClassDB::bind_method(D_METHOD("get_multiplayer", "for_path"), &SceneTree::get_multiplayer, DEFVAL(NodePath()));
	ClassDB::bind_method(D_METHOD("set_multiplayer_poll_enabled", "enabled"), &SceneTree::set_multiplayer_poll_enabled);
	ClassDB::bind_method(D_METHOD("is_multiplayer_poll_enabled"), &SceneTree::is_multiplayer_poll_enabled);
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "auto_accept_quit"), "set_auto_accept_quit", "is_auto_accept_quit");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "quit_on_go_back"), "set_quit_on_go_back", "is_quit_on_go_back");