### 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.
root->set_sdf_scale(sdf_scale);
#ifndef _3D_DISABLED
	{ // Load default fallback environment.
		// Get possible extensions.
		List<String> exts;
		ResourceLoader::get_recognized_extensions_for_type("Environment", &exts);
		String ext_hint;
		for (const String &E : exts) {
			if (!ext_hint.is_empty()) {
				ext_hint += ",";
			}
			ext_hint += "*." + E;
		}
		// Get path.
		String env_path = GLOBAL_DEF(PropertyInfo(Variant::STRING, "rendering/environment/defaults/default_environment", PROPERTY_HINT_FILE, ext_hint), "");
		// Setup property.
		env_path = env_path.strip_edges();
		if (!env_path.is_empty()) {
			Ref<Environment> env = ResourceLoader::load(env_path);
			if (env.is_valid()) {
				root->get_world_3d()->set_fallback_environment(env);
			} else {
				if (Engine::get_singleton()->is_editor_hint()) {
					// Fil
					ProjectSettings::get_singleton()->set("rendering/environment/defaults/default_environment", "");
				} else {
					// Fil
					ERR_PRINT(RTR("Default Environment as specified in the project setting \"rendering/environment/defaults/default_environment\" could not be loaded."));
				}
			}