### 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)

nodes_removed_on_group_call_lock++;
	}
	Vector<ObjectID> no_context_node_ids; 
	for (int i = gr_node_count - 1; i >= 0; i--) {
		if (p_viewport->is_input_handled()) {
			break;
		}
		Node *n = gr_nodes[i];
		if (nodes_removed_on_group_call.has(n)) {
			continue;
		}
		if (!n->can_process()) {
			continue;
		}
		switch (p_call_type) {
			case CALL_INPUT_TYPE_INPUT:
				n->_call_input(p_input);
				break;
			case CALL_INPUT_TYPE_SHORTCUT_INPUT: {
				const Control *c = Object::cast_to<Control>(n);
				if (c) {
					
					
					if (c->get_shortcut_context() == nullptr) {
						no_context_node_ids.append(n->get_instance_id());
						continue;
					}
					if (!c->is_focus_owner_in_shortcut_context()) {
						continue;
					}