### 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)

i--;
					pg_count--;
				}
			}
			if (pg_count != process_groups.size()) {
				process_groups.resize(pg_count);
			}
		}
		{
			
			process_groups.sort_custom<ProcessGroupSort>();
		}
		process_groups_dirty = false;
	}
	
	
	
	uint32_t group_count = process_groups.size();
	if (group_count == 0) {
		return;
	}
	process_last_pass++; 
	uint32_t from = 0;
	uint32_t process_count = 0;
	nodes_removed_on_group_call_lock++;
	int current_order = process_groups[0]->owner ? process_groups[0]->owner->data.process_thread_group_order : 0;
	bool current_threaded = process_groups[0]->owner ? process_groups[0]->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD : false;
	for (uint32_t i = 0; i <= group_count; i++) {
		int order = i < group_count && process_groups[i]->owner ? process_groups[i]->owner->data.process_thread_group_order : 0;