### 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.
ADD_PROPERTY(PropertyInfo(Variant::BOOL, "debug_collisions_hint"), "set_debug_collisions_hint", "is_debugging_collisions_hint");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "debug_paths_hint"), "set_debug_paths_hint", "is_debugging_paths_hint");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "debug_navigation_hint"), "set_debug_navigation_hint", "is_debugging_navigation_hint");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "paused"), "set_pause", "is_paused");
	ADD_PROPERTY(PropertyInfo(Variant::OBJECT, "edited_scene_root", PROPERTY_HINT_RESOURCE_TYPE, "Node", PROPERTY_USAGE_NONE), "set_edited_scene_root", "get_edited_scene_root");
	ADD_PROPERTY(PropertyInfo(Variant::OBJECT, "current_scene", PROPERTY_HINT_RESOURCE_TYPE, "Node", PROPERTY_USAGE_NONE), "set_current_scene", "get_current_scene");
	ADD_PROPERTY(PropertyInfo(Variant::OBJECT, "root", PROPERTY_HINT_RESOURCE_TYPE, "Node", PROPERTY_USAGE_NONE), "", "get_root");
	ADD_PROPERTY(PropertyInfo(Variant::BOOL, "multiplayer_poll"), "set_multiplayer_poll_enabled", "is_multiplayer_poll_enabled");
	ADD_SIGNAL(MethodInfo("tree_changed"));
	ADD_SIGNAL(MethodInfo("tree_process_mode_changed")); 
	ADD_SIGNAL(MethodInfo("node_added", PropertyInfo(Variant::OBJECT, "node", PROPERTY_HINT_RESOURCE_TYPE, "Node")));
	ADD_SIGNAL(MethodInfo("node_removed", PropertyInfo(Variant::OBJECT, "node", PROPERTY_HINT_RESOURCE_TYPE, "Node")));
	ADD_SIGNAL(MethodInfo("node_renamed", PropertyInfo(Variant::OBJECT, "node", PROPERTY_HINT_RESOURCE_TYPE, "Node")));
	ADD_SIGNAL(MethodInfo("node_configuration_warning_changed", PropertyInfo(Variant::OBJECT, "node", PROPERTY_HINT_RESOURCE_TYPE, "Node")));
	ADD_SIGNAL(MethodInfo("process_frame"));
	ADD_SIGNAL(MethodInfo("physics_frame"));
	BIND_ENUM_CONSTANT(GROUP_CALL_DEFAULT);
	BIND_ENUM_CONSTANT(GROUP_CALL_REVERSE);
	BIND_ENUM_CONSTANT(GROUP_CALL_DEFERRED);
	BIND_ENUM_CONSTANT(GROUP_CALL_UNIQUE);
}
SceneTree *SceneTree::singleton = nullptr;
SceneTree::IdleCallback SceneTree::idle_callbacks[SceneTree::MAX_IDLE_CALLBACKS];
int SceneTree::idle_callback_count = 0;
void SceneTree::_call_idle_callbacks() {
	for (int i = 0; i < idle_callback_count; i++) {
		idle_callbacks[i]();
	}
}
void SceneTree::add_idle_callback(IdleCallback p_callback) {