### 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)

return;
		}
		if (p_call_flags & GROUP_CALL_UNIQUE && p_call_flags & GROUP_CALL_DEFERRED) {
			ERR_FAIL_COND(ugc_locked);
			UGCall ug;
			ug.call = p_function;
			ug.group = p_group;
			if (unique_group_calls.has(ug)) {
				return;
			}
			Vector<Variant> args;
			for (int i = 0; i < p_argcount; i++) {
				args.push_back(*p_args[i]);
			}
			unique_group_calls[ug] = args;
			return;
		}
		_update_group_order(g);
		nodes_copy = g.nodes;
	}
	Node **gr_nodes = nodes_copy.ptrw();
	int gr_node_count = nodes_copy.size();
	{
		_THREAD_SAFE_METHOD_
		nodes_removed_on_group_call_lock++;
	}
	if (p_call_flags & GROUP_CALL_REVERSE) {
		for (int i = gr_node_count - 1; i >= 0; i--) {
			if (nodes_removed_on_group_call_lock && nodes_removed_on_group_call.has(gr_nodes[i])) {
				continue;