### 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.
Input *id = Input::get_singleton();
	if (id) {
		id->ensure_touch_mouse_raised();
	}
}
void SceneTree::_notification(int p_notification) {
	switch (p_notification) {
		case NOTIFICATION_TRANSLATION_CHANGED: {
			if (!Engine::get_singleton()->is_editor_hint()) {
				get_root()->propagate_notification(p_notification);
			}
		} break;
		case NOTIFICATION_OS_MEMORY_WARNING:
		case NOTIFICATION_OS_IME_UPDATE:
		case NOTIFICATION_WM_ABOUT:
		case NOTIFICATION_CRASH:
		case NOTIFICATION_APPLICATION_RESUMED:
		case NOTIFICATION_APPLICATION_PAUSED:
		case NOTIFICATION_APPLICATION_FOCUS_IN:
		case NOTIFICATION_APPLICATION_FOCUS_OUT: {
			
			get_root()->propagate_notification(p_notification);
		} break;
	}
}
bool SceneTree::is_auto_accept_quit() const {
	return accept_quit;
}
void SceneTree::set_auto_accept_quit(bool p_enable) {