### 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)

List<Connection> signal_connections;
	get_all_signal_connections(&signal_connections);
	for (const Connection &connection : signal_connections) {
		disconnect(connection.signal.get_name(), connection.callable);
	}
}
SceneTreeTimer::SceneTreeTimer() {}
void SceneTree::tree_changed() {
	tree_version++;
	emit_signal(tree_changed_name);
}
void SceneTree::node_added(Node *p_node) {
	emit_signal(node_added_name, p_node);
}
void SceneTree::node_removed(Node *p_node) {
	
	if (current_scene == p_node) {
		current_scene = nullptr;
	}
	emit_signal(node_removed_name, p_node);
	if (nodes_removed_on_group_call_lock) {
		nodes_removed_on_group_call.insert(p_node);
	}
}
void SceneTree::node_renamed(Node *p_node) {
	emit_signal(node_renamed_name, p_node);
}
SceneTree::Group *SceneTree::add_to_group(const StringName &p_group, Node *p_node) {