### 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)

}
	}
#endif 
	root->set_physics_object_picking(GLOBAL_DEF("physics/common/enable_object_picking", true));
	root->connect("close_requested", callable_mp(this, &SceneTree::_main_window_close));
	root->connect("go_back_requested", callable_mp(this, &SceneTree::_main_window_go_back));
	root->connect("focus_entered", callable_mp(this, &SceneTree::_main_window_focus_in));
#ifdef TOOLS_ENABLED
	edited_scene_root = nullptr;
#endif
	process_groups.push_back(&default_process_group);
}
SceneTree::~SceneTree() {
	if (root) {
		root->_set_tree(nullptr);
		root->_propagate_after_exit_tree();
		memdelete(root);
	}
	// Process groups are not deleted immediately, they may remain around. Delete them now.
	for (uint32_t i = 0; i < process_groups.size(); i++) {
		if (process_groups[i] != &default_process_group) {
			memdelete(process_groups[i]);
		}
	}
	memdelete(process_group_call_queue_allocator);
	if (singleton == this) {
		singleton = nullptr;
	}
}