### Godot version

v4.1.beta1.mono.official [828ec2c5d]

### System information

Godot v4.1.beta1.mono - Windows 10.0.22621 - Vulkan (Forward+) - dedicated NVIDIA GeForce RTX 4070 Ti (NVIDIA; 31.0.15.3598) - 13th Gen Intel(R) Core(TM) i5-13600K (20 Threads)

### Issue description

After the recent refactoring (https://github.com/godotengine/godot/pull/75901), Node process priority seems to be unimplemented. `ComparatorWithPriority` is no longer used anywhere in the engine code.

Nodes behave as if they all had the same priority, and running with `--single-threaded-scene` makes no difference.

### Steps to reproduce

Run the MRP, look at the order in which the nodes print their names.

In Godot 4.0.3: First, Default, Last
In Godot 4.1-beta1: Default, First, Last

### Minimal reproduction project

[ProcessPriorityBug.zip](https://github.com/godotengine/godot/files/11693028/ProcessPriorityBug.zip)

_THREAD_SAFE_METHOD_
	ERR_FAIL_NULL(p_object);
	p_object->_is_queued_for_deletion = true;
	delete_queue.push_back(p_object->get_instance_id());
}
int SceneTree::get_node_count() const {
	return nodes_in_tree_count;
}
void SceneTree::set_edited_scene_root(Node *p_node) {
#ifdef TOOLS_ENABLED
	edited_scene_root = p_node;
#endif
}
Node *SceneTree::get_edited_scene_root() const {
#ifdef TOOLS_ENABLED
	return edited_scene_root;
#else
	return nullptr;
#endif
}
void SceneTree::set_current_scene(Node *p_scene) {
	ERR_FAIL_COND_MSG(!Thread::is_main_thread(), "Changing scene can only be done from the main thread.");
	ERR_FAIL_COND(p_scene && p_scene->get_parent() != root);
	current_scene = p_scene;
}
Node *SceneTree::get_current_scene() const {
	return current_scene;
}
void SceneTree::_change_scene(Node *p_to) {