### 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.
ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Multiplayer can only be manipulated from the main thread.");
	multiplayer_poll = p_enabled;
}
bool SceneTree::is_multiplayer_poll_enabled() const {
	return multiplayer_poll;
}
void SceneTree::_bind_methods() {
	ClassDB::bind_method(D_METHOD("get_root"), &SceneTree::get_root);
	ClassDB::bind_method(D_METHOD("has_group", "name"), &SceneTree::has_group);
	ClassDB::bind_method(D_METHOD("is_auto_accept_quit"), &SceneTree::is_auto_accept_quit);
	ClassDB::bind_method(D_METHOD("set_auto_accept_quit", "enabled"), &SceneTree::set_auto_accept_quit);
	ClassDB::bind_method(D_METHOD("is_quit_on_go_back"), &SceneTree::is_quit_on_go_back);
	ClassDB::bind_method(D_METHOD("set_quit_on_go_back", "enabled"), &SceneTree::set_quit_on_go_back);
	ClassDB::bind_method(D_METHOD("set_debug_collisions_hint", "enable"), &SceneTree::set_debug_collisions_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_collisions_hint"), &SceneTree::is_debugging_collisions_hint);
	ClassDB::bind_method(D_METHOD("set_debug_paths_hint", "enable"), &SceneTree::set_debug_paths_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_paths_hint"), &SceneTree::is_debugging_paths_hint);
	ClassDB::bind_method(D_METHOD("set_debug_navigation_hint", "enable"), &SceneTree::set_debug_navigation_hint);
	ClassDB::bind_method(D_METHOD("is_debugging_navigation_hint"), &SceneTree::is_debugging_navigation_hint);
	ClassDB::bind_method(D_METHOD("set_edited_scene_root", "scene"), &SceneTree::set_edited_scene_root);
	ClassDB::bind_method(D_METHOD("get_edited_scene_root"), &SceneTree::get_edited_scene_root);
	ClassDB::bind_method(D_METHOD("set_pause", "enable"), &SceneTree::set_pause);
	ClassDB::bind_method(D_METHOD("is_paused"), &SceneTree::is_paused);
	ClassDB::bind_method(D_METHOD("create_timer", "time_sec", "process_always", "process_in_physics", "ignore_time_scale"), &SceneTree::create_timer, DEFVAL(true), DEFVAL(false), DEFVAL(false));
	ClassDB::bind_method(D_METHOD("create_tween"), &SceneTree::create_tween);
	ClassDB::bind_method(D_METHOD("get_processed_tweens"), &SceneTree::get_processed_tweens);
	ClassDB::bind_method(D_METHOD("get_node_count"), &SceneTree::get_node_count);
	ClassDB::bind_method(D_METHOD("get_frame"), &SceneTree::get_frame);
	ClassDB::bind_method(D_METHOD("quit", "exit_code"), &SceneTree::quit, DEFVAL(EXIT_SUCCESS));
	ClassDB::bind_method(D_METHOD("queue_delete", "obj"), &SceneTree::queue_delete);