### 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 (n->is_physics_processing_internal()) {
				n->notification(Node::NOTIFICATION_INTERNAL_PHYSICS_PROCESS);
			}
		} else {
			if (n->is_processing()) {
				n->notification(Node::NOTIFICATION_PROCESS);
			}
			if (n->is_processing_internal()) {
				n->notification(Node::NOTIFICATION_INTERNAL_PROCESS);
			}
		}
	}
	p_group->call_queue.flush(); 
}
void SceneTree::_process_groups_thread(uint32_t p_index, bool p_physics) {
	Node::current_process_thread_group = local_process_group_cache[p_index]->owner;
	_process_group(local_process_group_cache[p_index], p_physics);
	Node::current_process_thread_group = nullptr;
}
void SceneTree::_process(bool p_physics) {
	if (process_groups_dirty) {
		{
			
			
			ProcessGroup **pg_ptr = (ProcessGroup **)process_groups.ptr(); 
			uint32_t pg_count = process_groups.size();
			for (uint32_t i = 0; i < pg_count; i++) {
				if (pg_ptr[i]->removed) {
					
					pg_ptr[i] = pg_ptr[pg_count - 1];