### 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)

bool threaded = i < group_count && process_groups[i]->owner ? process_groups[i]->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD : false;
		if (i == group_count || current_order != order || current_threaded != threaded) {
			if (process_count > 0) {
				
				bool using_threads = process_groups[from]->owner && process_groups[from]->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD && !node_threading_disabled;
				if (using_threads) {
					local_process_group_cache.clear();
				}
				for (uint32_t j = from; j < i; j++) {
					if (process_groups[j]->last_pass == process_last_pass) {
						if (using_threads) {
							local_process_group_cache.push_back(process_groups[j]);
						} else {
							_process_group(process_groups[j], p_physics);
						}
					}
				}
				if (using_threads) {
					WorkerThreadPool::GroupID id = WorkerThreadPool::get_singleton()->add_template_group_task(this, &SceneTree::_process_groups_thread, p_physics, local_process_group_cache.size(), -1, true);
					WorkerThreadPool::get_singleton()->wait_for_group_task_completion(id);
				}
			}
			if (i == group_count) {
				
				break;
			}
			from = i;
			current_threaded = threaded;
			current_order = order;
		}