### 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)

_THREAD_SAFE_METHOD_
	
	List<Ref<Tween>>::Element *L = tweens.back();
	for (List<Ref<Tween>>::Element *E = tweens.front(); E;) {
		List<Ref<Tween>>::Element *N = E->next();
		
		if (!E->get()->can_process(paused) || (p_physics == (E->get()->get_process_mode() == Tween::TWEEN_PROCESS_IDLE))) {
			if (E == L) {
				break;
			}
			E = N;
			continue;
		}
		if (!E->get()->step(p_delta)) {
			E->get()->clear();
			tweens.erase(E);
		}
		if (E == L) {
			break;
		}
		E = N;
	}
}
void SceneTree::finalize() {