### 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_V_MSG(!Thread::is_main_thread(), Ref<MultiplayerAPI>(), "Multiplayer can only be manipulated from the main thread.");
	Ref<MultiplayerAPI> out = multiplayer;
	for (const KeyValue<NodePath, Ref<MultiplayerAPI>> &E : custom_multiplayers) {
		const Vector<StringName> snames = E.key.get_names();
		const Vector<StringName> tnames = p_for_path.get_names();
		if (tnames.size() < snames.size()) {
			continue;
		}
		const StringName *sptr = snames.ptr();
		const StringName *nptr = tnames.ptr();
		bool valid = true;
		for (int i = 0; i < snames.size(); i++) {
			if (sptr[i] != nptr[i]) {
				valid = false;
				break;
			}
		}
		if (valid) {
			out = E.value;
			break;
		}
	}
	return out;
}
void SceneTree::set_multiplayer(Ref<MultiplayerAPI> p_multiplayer, const NodePath &p_root_path) {