### 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)

if (!g.changed) {
		return;
	}
	if (g.nodes.is_empty()) {
		return;
	}
	Node **gr_nodes = g.nodes.ptrw();
	int gr_node_count = g.nodes.size();
	SortArray<Node *, Node::Comparator> node_sort;
	node_sort.sort(gr_nodes, gr_node_count);
	g.changed = false;
}
void SceneTree::call_group_flagsp(uint32_t p_call_flags, const StringName &p_group, const StringName &p_function, const Variant **p_args, int p_argcount) {
	Vector<Node *> nodes_copy;
	{
		_THREAD_SAFE_METHOD_
		HashMap<StringName, Group>::Iterator E = group_map.find(p_group);
		if (!E) {
			return;
		}
		Group &g = E->value;
		if (g.nodes.is_empty()) {