### 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.");
	if (p_root_path.is_empty()) {
		ERR_FAIL_COND(!p_multiplayer.is_valid());
		if (multiplayer.is_valid()) {
			multiplayer->object_configuration_remove(nullptr, NodePath("/" + root->get_name()));
		}
		multiplayer = p_multiplayer;
		multiplayer->object_configuration_add(nullptr, NodePath("/" + root->get_name()));
	} else {
		if (custom_multiplayers.has(p_root_path)) {
			custom_multiplayers[p_root_path]->object_configuration_remove(nullptr, p_root_path);
		}
		if (p_multiplayer.is_valid()) {
			custom_multiplayers[p_root_path] = p_multiplayer;
			p_multiplayer->object_configuration_add(nullptr, p_root_path);
		}
	}
}
void SceneTree::set_multiplayer_poll_enabled(bool p_enabled) {