### 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.
_call_idle_callbacks();
#ifdef TOOLS_ENABLED
#ifndef _3D_DISABLED
	if (Engine::get_singleton()->is_editor_hint()) {
		
		String env_path = GLOBAL_GET(SNAME("rendering/environment/defaults/default_environment"));
		env_path = env_path.strip_edges(); 
		String cpath;
		Ref<Environment> fallback = get_root()->get_world_3d()->get_fallback_environment();
		if (fallback.is_valid()) {
			cpath = fallback->get_path();
		}
		if (cpath != env_path) {
			if (!env_path.is_empty()) {
				fallback = ResourceLoader::load(env_path);
				if (fallback.is_null()) {
					
					ProjectSettings::get_singleton()->set("rendering/environment/defaults/default_environment", "");
				}
			} else {
				fallback.unref();
			}
			get_root()->get_world_3d()->set_fallback_environment(fallback);
		}
	}
#endif 
#endif 
	return _quit;
}
void SceneTree::process_timers(double p_delta, bool p_physics_frame) {